#72231978

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₈ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₈
Frattini, cosocle	C₄×C₈	C₂⁵
socle	C₂²	C₂³×C₄⋊C₈
prime radical	#3444	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ #60204 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2444 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(3,9)(4,11)(7,14)(8,13)(10,15)(12,16)(17,18,20,25,21,26,30,38,22,24,31,36,32,37,42,46)(19,23,28,33,29,34,41,43,27,35,39,44,40,45,47,48);(1,2)(3,7)(4,8)(5,6)(9,14)(10,12)(11,13)(15,16)(17,19)(18,23)(20,28)(21,29)(22,27)(24,35)(25,33)(26,34)(30,41)(31,39)(32,40)(36,44)(37,45)(38,43)(42,47)(46,48);(18,24)(19,27)(25,36)(26,37)(28,39)(29,40)(38,46)(41,47);(2,6)(7,12)(8,13)(14,16);(1,3,4,9,5,10,11,15)(2,7,8,14,6,12,13,16)

Polycyclic presentation:

2^10|a^2=f,e^2=g,f^2=h,g^2=i,h^2=j,c^a=cj,c^b=cj,d^b=di,e^a=eg,g^a=gi

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72231984
maximal normal subgroup of	#72231984

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