#6506080

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

1389 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)(17,18,22,28,23,29,21,27)(19,24,31,40,32,41,36,46)(20,26,34,43,35,45,37,25)(30,39,48,42,33,44,47,38);(1,3,5,4)(2,6,7,8)(9,10)(11,15)(12,14)(13,16)(17,19,23,32)(18,24,29,41)(20,33,35,30)(21,36,22,31)(25,42,43,38)(26,44,45,39)(27,46,28,40)(34,47,37,48);(9,11)(10,13)(12,16)(14,15)(17,20,23,35)(18,25,29,43)(19,30,32,33)(21,34,22,37)(24,38,41,42)(26,28,45,27)(31,47,36,48)(39,40,44,46);(11,16)(13,15)(17,20)(18,26)(19,30)(21,37)(22,34)(23,35)(24,39)(25,27)(28,43)(29,45)(31,48)(32,33)(36,47)(38,46)(40,42)(41,44);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(17,21,23,22)(18,27,29,28)(19,31,32,36)(20,34,35,37)(24,40,41,46)(25,26,43,45)(30,47,33,48)(38,44,42,39)

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,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, 6413276);