#6771323

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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	C₂×#121	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1501 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=i,d^2=fhi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6678519);