#6543959

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1425 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)(19,28)(20,33)(21,27)(22,36)(23,26)(24,32)(25,34)(29,38)(30,42)(31,39)(35,41)(37,40)(43,45)(44,48)(46,47);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,30)(18,24,26,39)(20,29,35,44)(21,32,22,31)(25,38,40,48)(27,42,36,28)(33,43,41,47)(34,46,37,45);(10,13)(14,16)(17,20)(18,25)(19,29)(21,34)(22,37)(23,35)(24,38)(26,40)(27,41)(28,43)(30,44)(31,45)(32,46)(33,36)(39,48)(42,47);(1,3,5,4)(2,7,6,8)(9,11)(10,14)(12,15)(13,16)(18,26)(19,30)(25,40)(27,36)(29,44)(31,32)(33,41)(45,46);(1,4,5,3)(2,8,6,7)(11,15)(14,16)(17,21,23,22)(18,27,26,36)(19,31,30,32)(20,34,35,37)(24,28,39,42)(25,41,40,33)(29,45,44,46)(38,43,48,47)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6451155);