#6520889

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500456
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 ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1398 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)(17,18,22,26)(19,27,28,40)(20,24,31,38)(21,33,34,25)(23,35,36,30)(29,41,42,48)(32,45,44,39)(37,46,47,43);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(19,28)(23,36)(27,40)(29,42)(30,35)(37,47)(41,48)(43,46);(10,13)(11,15)(17,19,21,30,22,28,34,35)(18,23,25,27,26,36,33,40)(20,29,32,43,31,42,44,46)(24,37,39,41,38,47,45,48);(9,11,12,15)(10,14,13,16)(17,20,22,31)(18,24,26,38)(19,29,28,42)(21,32,34,44)(23,37,36,47)(25,39,33,45)(27,41,40,48)(30,43,35,46);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(20,31)(24,38)(29,42)(32,44)(37,47)(39,45)(41,48)(43,46)

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=h,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ehi,d^c=dhi,e^d=ei

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, 6428085);