#6506327

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500186
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

1391 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)(3,8)(4,6)(5,7)(17,18,23,29,22,28,21,27)(19,30,32,46,33,24,38,42)(20,34,36,44,37,26,39,25)(31,41,48,40,35,45,47,43);(1,3,5,4)(2,6,7,8)(9,10)(11,15)(12,14)(13,16)(17,19,22,33)(18,24,28,30)(20,35,37,31)(21,38,23,32)(25,43,44,40)(26,45,34,41)(27,46,29,42)(36,47,39,48);(9,11)(10,13)(12,16)(14,15)(17,20,22,37)(18,25,28,44)(19,31,33,35)(21,36,23,39)(24,40,30,43)(26,27,34,29)(32,47,38,48)(41,46,45,42);(11,16)(13,15)(17,20)(18,26)(19,31)(21,39)(22,37)(23,36)(24,41)(25,29)(27,44)(28,34)(30,45)(32,48)(33,35)(38,47)(40,42)(43,46);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(17,21,22,23)(18,27,28,29)(19,32,33,38)(20,36,37,39)(24,42,30,46)(25,34,44,26)(31,47,35,48)(40,41,43,45)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6413523);