#6820338

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1322 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)(9,10,13,15)(11,14,12,16)(17,18,22,26)(19,23,29,35)(20,24,31,37)(21,25,32,38)(27,33,40,44)(28,34,41,45)(30,36,42,46)(39,43,47,48);(1,2)(3,6)(4,5)(7,8)(9,11,13,12)(10,14,15,16)(19,27)(20,30)(21,32)(23,33)(24,36)(25,38)(28,41)(29,40)(31,42)(34,45)(35,44)(37,46);(9,12,13,11)(10,14,15,16)(17,19,21,27,22,29,32,40)(18,23,25,33,26,35,38,44)(20,28,30,39,31,41,42,47)(24,34,36,43,37,45,46,48);(1,3,4,7)(2,6,5,8)(20,31)(24,37)(28,41)(30,42)(34,45)(36,46)(39,47)(43,48);(2,5)(6,8)(17,20,21,30,22,31,32,42)(18,24,25,36,26,37,38,46)(19,28,27,39,29,41,40,47)(23,34,33,43,35,45,44,48)

Polycyclic presentation:

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

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, 6727534);