#6560154

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535815
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1599 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:

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

Polycyclic presentation:

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

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, 6467350);