#72237795

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	#1269009
commutator, Frattini, cosocle	C₂⁵	C₂⁵
prime radical	#394064	C₂
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

5283 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:

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

Polycyclic presentation:

2^10|c^2=ij,e^2=hj,b^a=bf,c^a=ci,c^b=ch,d^a=dij,d^b=dg,d^c=dij,e^c=eghj,e^d=eij,f^c=fj,f^d=fj,g^c=gj,g^e=gj,h^c=hj,i^b=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237804, #72240477, #72240493, …
maximal normal subgroup of	#72237804, #72240477, #72240493, …

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, 3, F.9*F.10);;
gap> SetPower(coll, 5, F.8*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.8*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, 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, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);