#6562212

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1452 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)(3,7)(4,6)(5,8)(17,18)(19,28)(20,33)(21,26)(22,37)(23,27)(24,32)(25,35)(29,38)(30,41)(31,39)(34,40)(36,42)(43,46)(44,47)(45,48);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,29)(21,32)(22,31)(23,30)(25,38)(26,41)(27,39)(28,37)(33,43)(34,46)(35,45)(36,44)(40,47)(42,48);(1,3,5,4)(2,7,8,6)(10,13)(14,16)(17,20,23,36)(18,25,27,40)(19,29,30,44)(21,35,22,34)(24,38,39,47)(26,42,37,33)(28,43,41,48)(31,46,32,45);(1,4,5,3)(2,6,8,7)(9,11)(10,14)(12,15)(13,16)(19,30)(24,39)(28,41)(29,44)(31,32)(38,47)(43,48)(45,46);(11,15)(14,16)(17,21,23,22)(18,26,27,37)(19,31,30,32)(20,34,36,35)(24,28,39,41)(25,33,40,42)(29,45,44,46)(38,48,47,43)

Polycyclic presentation:

2^9|b^2=fgi,c^2=fg,d^2=fgi,e^2=i,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,f^b=fi,g^b=gi,e^c=ei,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, 6469408);