#72233801

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#15257
commutator, Frattini, cosocle	C₂⁴×C₄	C₂⁴
LCS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

4111 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^10|a^2=g,f^2=i,b^a=bgh,c^a=cf,c^b=ce,d^a=dg,d^b=dh,d^c=di,e^a=ei,e^d=ej,f^b=fj,f^c=fi,f^d=fi,g^c=gi,h^c=hj

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72233808
maximal normal subgroup of	#72233808

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