#6699993

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	#36018	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1728 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)(11,15)(12,14)(13,16)(17,18,22,26)(19,23,29,38)(20,24,32,41)(21,25,33,42)(27,36,34,43)(28,37,31,40)(30,39,35,44)(45,47,46,48);(1,2,4,5)(3,6,7,8)(9,11,12,16)(10,13,14,15)(19,27)(20,30)(21,33)(23,36)(24,39)(25,42)(29,34)(32,35)(38,43)(41,44)(45,46)(47,48);(10,14)(13,15)(17,19,22,29)(18,23,26,38)(20,31,32,28)(21,34,33,27)(24,40,41,37)(25,43,42,36)(30,46,35,45)(39,48,44,47);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(20,32)(24,41)(28,31)(30,35)(37,40)(39,44)(45,46)(47,48);(2,5)(6,8)(11,16)(13,15)(17,20,22,32)(18,24,26,41)(19,28,29,31)(21,35,33,30)(23,37,38,40)(25,44,42,39)(27,45,34,46)(36,47,43,48)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=fg,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,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, 6607189);