#6856978

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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 ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1308 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)(3,6)(4,8)(5,7)(9,10,13,15)(11,14,12,16)(17,18,22,27)(19,23,28,36)(20,25,31,39)(21,24,33,26)(29,35,41,46)(30,37,42,34)(32,38,44,40)(43,47,48,45);(1,3,6,2)(4,8,5,7)(9,10)(11,16)(12,14)(13,15)(17,19)(18,23)(20,29)(21,30)(22,28)(24,37)(25,35)(26,34)(27,36)(31,41)(32,43)(33,42)(38,47)(39,46)(40,45)(44,48);(9,11,13,12)(10,14,15,16)(18,24)(19,28)(21,33)(23,34)(25,38)(26,27)(29,41)(32,44)(35,45)(36,37)(39,40)(46,47);(1,4,6,5)(2,7,3,8)(9,12,13,11)(10,14,15,16)(17,20)(18,25)(19,29)(21,32)(22,31)(23,35)(24,38)(26,40)(27,39)(28,41)(30,43)(33,44)(34,45)(36,46)(37,47)(42,48);(1,5,6,4)(2,7,3,8)(19,28)(20,31)(23,36)(25,39)(30,42)(32,44)(34,37)(38,40)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6764174);