#72316749

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#2021
commutator, Frattini, cosocle	C₂×C₄×D₄	C₂⁴
LCS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

3515 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^10|a^2=g,b^2=j,c^2=j,f^2=i,b^a=bgi,c^a=ci,d^a=de,d^b=defi,d^c=di,e^a=eh,f^b=fi,f^d=fi,g^d=gh,g^e=gj,h^a=hj,h^b=hj

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72316767, #72316802, #72316810
maximal normal subgroup of	#72316767, #72316802, #72316810

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