#72285430

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	C₂²	#46996
commutator, Frattini, cosocle	C₂⁵	C₂⁵
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

3371 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^10|b^2=ij,c^2=fgh,d^2=ghij,e^2=j,b^a=bij,c^a=cf,c^b=cfg,d^a=dgi,d^b=dfi,d^c=dhij,e^b=efi,e^c=efi,e^d=ehij,f^c=fj,f^d=fi,g^c=gij,g^d=gj,g^e=gj,h^a=hj,h^b=hi,h^c=hi,h^d=hj

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285526, #72310874, #72338769, …
maximal normal subgroup of	#72285526, #72310874, #72338769, …

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