#6820112

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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1397 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=ghi,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,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, 6727308);