#6501344

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499823
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	#23854	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1755 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)(17,18,23,28)(19,24,32,42)(20,26,34,43)(21,27,22,37)(25,36,44,33)(29,45,30,39)(31,40,35,41)(38,46,48,47);(2,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,19,23,32)(18,24,28,42)(20,29,34,30)(21,35,22,31)(25,38,44,48)(26,45,43,39)(27,41,37,40)(33,47,36,46);(9,11)(10,13)(12,16)(14,15)(17,20)(18,25)(19,29)(21,33)(22,36)(23,34)(24,38)(26,37)(27,43)(28,44)(30,32)(31,46)(35,47)(39,41)(40,45)(42,48);(1,3,4,7)(2,6,5,8)(11,16)(13,15)(17,20)(18,26)(19,30)(21,36)(22,33)(23,34)(24,39)(25,37)(27,44)(28,43)(29,32)(31,46)(35,47)(38,41)(40,48)(42,45);(10,14)(13,15)(17,21,23,22)(18,27,28,37)(19,31,32,35)(20,33,34,36)(24,40,42,41)(25,43,44,26)(29,46,30,47)(38,45,48,39)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,g^c=gi,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, 6408540);