#6828722

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

1436 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18,23,28)(19,24,31,39)(20,25,30,38)(21,26,35,43)(22,27,32,40)(29,37,46,48)(33,41,34,42)(36,44,45,47);(9,11)(10,13)(12,16)(14,15)(17,19,23,31)(18,24,28,39)(20,32,30,22)(21,34,35,33)(25,40,38,27)(26,42,43,41)(29,45,46,36)(37,47,48,44);(2,5)(6,8)(10,14)(11,16)(17,20,22,19,23,30,32,31)(18,25,27,24,28,38,40,39)(21,34,36,46,35,33,45,29)(26,42,44,48,43,41,47,37);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(21,35)(26,43)(29,46)(33,34)(36,45)(37,48)(41,42)(44,47);(17,21,32,45,23,35,22,36)(18,26,40,47,28,43,27,44)(19,29,20,33,31,46,30,34)(24,37,25,41,39,48,38,42)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fh,d^2=fg,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,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, 6735918);