#6566293

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535655
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1699 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 52 points:

(1,2)(3,4)(5,6,9,12)(7,10,13,16)(8,11,15,18)(14,17,19,20)(21,22)(23,33)(24,38)(25,30)(26,31)(27,32)(28,35)(29,41)(34,42)(36,44)(37,43)(39,46)(40,45)(47,50)(48,52)(49,51);(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23,26,36,27,37,25,35)(22,28,30,43,32,44,31,33)(24,34,41,50,39,48,40,49)(29,42,38,51,45,52,46,47);(2,4)(7,13)(10,16)(14,19)(17,20)(21,24,25,40,27,39,26,41)(22,29,31,46,32,45,30,38)(23,34,35,49,37,48,36,50)(28,42,33,47,44,52,43,51);(5,8)(6,11)(7,14)(9,15)(10,17)(12,18)(13,19)(16,20)(24,39)(29,45)(34,48)(38,46)(40,41)(42,52)(47,51)(49,50);(8,15)(11,18)(14,19)(17,20)(21,25,27,26)(22,30,32,31)(23,35,37,36)(24,40,39,41)(28,43,44,33)(29,38,45,46)(34,49,48,50)(42,51,52,47)

Polycyclic presentation:

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

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, 6473489);