#6624751

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574015
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1402 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)(4,8)(5,6)(9,10)(11,15)(12,16)(13,14)(17,18,22,27)(19,28,31,40)(20,33,36,43)(21,25,29,26)(23,37,39,32)(24,41,44,35)(30,42,34,38)(45,48,47,46);(2,6)(7,8)(10,14)(11,12)(17,19,29,37,22,31,21,32)(18,23,25,40,27,39,26,28)(20,34,41,46,36,30,35,48)(24,42,43,45,44,38,33,47);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,15,14,16)(17,20,21,35,22,36,29,41)(18,24,26,33,27,44,25,43)(19,30,32,46,31,34,37,48)(23,38,28,45,39,42,40,47);(1,4,5,3)(2,8,6,7)(11,12)(15,16)(19,31)(23,39)(28,40)(30,34)(32,37)(38,42)(45,47)(46,48);(9,12,13,11)(10,16,14,15)(17,21,22,29)(18,25,27,26)(19,32,31,37)(20,35,36,41)(23,40,39,28)(24,43,44,33)(30,46,34,48)(38,47,42,45)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=f,d^2=fgh,e^2=h,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=ehi,d^c=dhi,e^d=ehi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6531947);