#72316723

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#2021
commutator, Frattini, cosocle	C₂×C₄×D₄	C₂⁴
prime radical	#6487648	C₂
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

3435 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:

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

Polycyclic presentation:

2^10|a^2=g,c^2=j,d^2=j,f^2=i,b^a=bg,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	#72316746, #72316779, #72316810
maximal normal subgroup of	#72316746, #72316779, #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, 3, F.10);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 6, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
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);