#6561469

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535823
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	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1315 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6468665);