#72240468

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#3514537
commutator, Frattini, cosocle	C₂⁵	C₂⁵
prime radical	#401060	C₂
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

4045 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 5 generators on 32 points:

(5,18)(7,11)(8,15)(9,19)(12,26)(21,23)(22,25)(24,28);(1,2)(3,11,14,7)(4,15)(5,19,18,9)(6,10)(8,17)(12,23,26,21)(13,22)(16,28,29,24)(20,25)(27,32)(30,31);(1,3,6,14)(2,7,10,11)(4,12,17,26)(5,20,18,13)(8,23,15,21)(9,22,19,25)(16,27,29,30)(24,31,28,32);(1,4,6,17)(2,8)(3,12)(5,16,18,29)(7,21,11,23)(9,24)(10,15)(13,27)(14,26)(19,28)(20,30)(22,31,25,32);(1,5,6,18)(2,9)(3,13,14,20)(4,16)(7,22)(8,24,15,28)(10,19)(11,25)(12,27)(17,29)(21,31,23,32)(26,30)

Polycyclic presentation:

2^10|b^2=f,c^2=j,d^2=ij,e^2=hij,b^a=bh,c^a=cg,c^b=cfi,d^a=dgh,d^b=dgi,d^c=dgj,e^a=ehj,e^b=egj,e^c=ehij,e^d=eghij,f^c=fj,f^e=fj,g^b=gj,g^d=gj,g^e=gj,h^c=hj,h^d=hj,i^b=ij,i^c=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240477, #72251302, #72334902, …
maximal normal subgroup of	#72240477, #72251302, #72334902, …

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 3, F.10);;
gap> SetPower(coll, 4, F.9*F.10);;
gap> SetPower(coll, 5, F.8*F.9*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);