#6720113

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1397 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=gi,c^2=fi,d^2=f,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6627309);