#6814117

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
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

1412 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)(4,8)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18)(19,27)(20,25)(21,23)(22,26)(24,29)(28,38)(30,42)(31,45)(32,39)(33,37)(34,36)(35,41)(40,43)(44,47)(46,48);(1,3,6,2)(4,8,5,7)(9,11,12,15)(10,13,14,16)(18,23)(19,28)(20,31)(22,33)(24,38)(25,41)(27,40)(29,43)(34,35)(36,45)(39,42)(44,46);(10,14)(13,16)(17,19,22,28,21,29,33,43)(18,24,26,40,23,27,37,38)(20,32,35,46,34,30,31,44)(25,42,41,47,36,39,45,48);(1,4,6,5)(2,7,3,8)(18,23)(19,29)(25,36)(26,37)(28,43)(30,32)(41,45)(44,46);(1,5,6,4)(2,7,3,8)(17,20,33,31,21,34,22,35)(18,25,37,45,23,36,26,41)(19,30,43,46,29,32,28,44)(24,39,38,47,27,42,40,48)

Polycyclic presentation:

2^9|b^2=fg,c^2=h,d^2=fi,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=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, 6721313);