#6533722

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1728 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)(19,28)(20,32)(21,26)(22,27)(23,37)(24,31)(25,35)(29,44)(30,40)(33,41)(34,42)(36,39)(38,47)(43,45)(46,48);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,29)(21,36)(22,31)(23,30)(25,38)(26,40)(27,28)(32,48)(33,47)(34,46)(35,45)(37,39)(41,43)(42,44);(10,13)(14,16)(17,20,22,34)(18,25,27,41)(19,29,31,46)(21,35,23,33)(24,38,28,43)(26,42,37,32)(30,47,36,45)(39,48,40,44);(1,3,4,7)(2,6,5,8)(9,11)(10,14)(12,15)(13,16);(11,15)(14,16)(17,21,22,23)(18,26,27,37)(19,30,31,36)(20,33,34,35)(24,39,28,40)(25,32,41,42)(29,45,46,47)(38,44,43,48)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6440918);