#6484566

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470161
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1556 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)(9,10)(11,15)(12,16)(13,14)(17,18,22,27,23,26,21,28)(19,29,32,24,33,41,31,40)(20,34,36,43,38,44,37,25)(30,42,46,48,35,39,47,45);(2,5)(6,8)(10,14)(11,12)(17,19,23,33)(18,24,26,40)(20,35,38,30)(21,32,22,31)(25,42,43,39)(27,29,28,41)(34,45,44,48)(36,46,37,47);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,15,14,16)(17,20)(18,25)(19,30)(21,36)(22,37)(23,38)(24,39)(26,43)(27,44)(28,34)(29,45)(31,46)(32,47)(33,35)(40,42)(41,48);(9,12,13,11)(10,16,14,15)(18,26)(24,40)(25,43)(27,28)(29,41)(34,44)(39,42)(45,48);(10,14)(15,16)(17,21,23,22)(18,27,26,28)(19,31,33,32)(20,36,38,37)(24,41,40,29)(25,44,43,34)(30,46,35,47)(39,48,42,45)

Polycyclic presentation:

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

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, 6391762);