#72225682

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 ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

9400 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 5 generators on 24 points:

(15,19)(17,23);(1,2)(3,7)(4,6)(5,8)(9,10)(11,14)(12,19,20,15)(13,16)(17,24,23,21)(18,22);(1,3,4,8)(2,5,6,7)(9,11,13,18)(10,14,16,22)(12,17,20,23)(15,21,19,24);(2,6)(3,8)(9,12)(10,15)(11,17)(13,20)(14,21)(16,19)(18,23)(22,24);(3,8)(5,7)(10,16)(11,18)(14,22)(15,19)(17,23)(21,24)

Polycyclic presentation:

2^10|b^2=f,c^2=gj,b^a=bf,c^a=cf,c^b=cg,d^a=di,d^b=dgh,d^c=dg,e^b=ehij,e^c=eghj,f^d=fj,h^b=hj,i^b=ij,i^c=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225688, #72225729, #72318468, …
maximal normal subgroup of	#72225688, #72225729, #72318468, …

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.7*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.8*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 8, 2, 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);