#6502328

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499804
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1603 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)(17,18,22,26)(19,23,28,37)(20,30,31,40)(21,33,35,25)(24,34,39,32)(27,41,43,48)(29,45,44,38)(36,46,47,42);(1,3,5,4)(2,6,7,8)(9,10)(11,15)(12,14)(13,16)(17,19)(18,23)(20,27)(21,29)(22,28)(24,36)(25,38)(26,37)(30,41)(31,43)(32,42)(33,45)(34,46)(35,44)(39,47)(40,48);(9,11)(10,13)(12,16)(14,15)(17,20,22,31)(18,24,26,39)(19,27,28,43)(21,34,35,32)(23,36,37,47)(25,40,33,30)(29,46,44,42)(38,48,45,41);(11,16)(13,15)(19,28)(20,31)(23,37)(24,39)(29,44)(30,40)(32,34)(38,45);(1,4,5,3)(2,6,7,8)(10,14)(13,15)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6409524);