#72237802

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#1100069
commutator, Frattini, cosocle	C₂⁵	C₂⁵
prime radical	#404015	C₂
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

4089 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:

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

Polycyclic presentation:

2^10|a^2=f,b^2=gi,e^2=h,b^a=bfg,c^a=ci,d^a=dh,d^b=dh,d^c=di,e^a=eh,e^b=ehij,e^c=egj,f^b=fj,f^c=fj,g^a=gj,g^b=gj,g^e=gj,h^c=hj,i^a=ij,i^b=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237804, #72237925, #72251313, …
maximal normal subgroup of	#72237804, #72237925, #72251313, …

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