#6534616

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×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1684 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)(3,6)(4,5)(7,8)(17,18)(19,29)(20,34)(21,26)(22,27)(23,28)(24,33)(25,36)(30,44)(31,40)(32,39)(35,42)(37,41)(38,45)(43,46)(47,48);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,30)(21,32)(22,33)(23,31)(25,38)(26,29)(27,40)(28,39)(34,48)(35,46)(36,47)(37,45)(41,43)(42,44);(10,13)(14,16)(17,20,23,37)(18,25,28,42)(19,30,31,45)(21,36,22,35)(24,38,39,44)(26,41,27,34)(29,43,40,48)(32,47,33,46);(1,3,4,7)(2,6,5,8)(9,11)(10,14)(12,15)(13,16)(19,31)(24,39)(29,40)(30,45)(32,33)(38,44)(43,48)(46,47);(11,15)(14,16)(17,21,23,22)(18,26,28,27)(19,32,31,33)(20,35,37,36)(24,29,39,40)(25,34,42,41)(30,46,45,47)(38,48,44,43)

Polycyclic presentation:

2^9|c^2=i,d^2=fg,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,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, 6441812);