#6484340

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470884
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×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1459 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)(9,10)(11,16)(12,14)(13,15)(17,18,22,26)(19,27,31,43)(20,24,32,39)(21,35,28,25)(23,36,38,30)(29,42,33,46)(34,48,44,41)(37,47,40,45);(2,6)(7,8)(11,12)(14,16)(20,32)(24,39)(29,33)(34,44)(37,40)(41,48)(42,46)(45,47);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,14,15,16)(17,19,28,36,22,31,21,30)(18,23,35,43,26,38,25,27)(20,33,44,45,32,29,34,47)(24,40,48,42,39,37,41,46);(9,12,13,11)(10,14,15,16)(17,20,22,32)(18,24,26,39)(19,29,31,33)(21,34,28,44)(23,37,38,40)(25,41,35,48)(27,42,43,46)(30,45,36,47);(1,4,5,3)(2,8,6,7)(10,15)(14,16)(20,32)(24,39)(29,33)(34,44)(37,40)(41,48)(42,46)(45,47)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6391536);