#6502358

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499823
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 ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1607 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)(19,30,34,25)(20,35,21,27)(22,28,24,39)(26,37,45,36)(31,41,32,46)(33,43,38,42)(40,48,44,47);(1,3,5,4)(2,6,7,8)(9,10)(11,15)(12,14)(13,16)(17,19,23,34)(18,25,29,30)(20,32,21,31)(22,38,24,33)(26,44,45,40)(27,46,35,41)(28,43,39,42)(36,47,37,48);(9,11)(10,13)(12,16)(14,15)(17,20,23,21)(18,26,29,45)(19,31,34,32)(22,36,24,37)(25,40,30,44)(27,39,35,28)(33,47,38,48)(41,42,46,43);(11,16)(13,15)(17,21,23,20)(18,27,29,35)(19,32,34,31)(22,36,24,37)(25,41,30,46)(26,39,45,28)(33,47,38,48)(40,42,44,43);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(17,22,23,24)(18,28,29,39)(19,33,34,38)(20,36,21,37)(25,42,30,43)(26,27,45,35)(31,47,32,48)(40,46,44,41)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^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,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, 6409554);