#6557647

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535664
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 ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1325 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=g,c^2=fi,d^2=fgi,e^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=ei,e^d=ehi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6464843);