#6503230

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500076
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×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1692 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=f,d^2=i,e^2=i,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

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, 6410426);