#6909272

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1463 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=gi,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfh,e^c=ei,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6816468);