#72249406

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#32898
commutator, Frattini, cosocle	C₂³×D₄	C₂⁴
LCS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#121 ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

4404 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^10|b^2=i,c^2=j,d^2=i,c^a=cf,c^b=cg,d^b=dij,d^c=de,e^a=ej,e^b=eh,f^b=fi,f^d=fj,g^a=gi,g^d=ghj,g^e=gj,h^c=hj

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249415, #72249432
maximal normal subgroup of	#72249415, #72249432
maximal quotient of	#72263329

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