#6679051

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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1696 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=i,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,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, 6586247);