#6566299

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

1703 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)(21,22,27,32)(23,33,37,43)(24,38,41,45)(25,30,26,31)(28,36,44,35)(29,40,46,39)(34,42,50,52)(47,49,51,48);(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)(24,34,39,48,41,50,40,49)(29,42,38,51,46,52,45,47);(2,4)(7,13)(10,16)(14,19)(17,20)(21,24,25,40,27,41,26,39)(22,29,31,45,32,46,30,38)(23,34,35,49,37,50,36,48)(28,42,33,47,44,52,43,51);(5,8)(6,11)(7,14)(9,15)(10,17)(12,18)(13,19)(16,20)(21,25,27,26)(22,30,32,31)(23,35,37,36)(24,39,41,40)(28,43,44,33)(29,45,46,38)(34,48,50,49)(42,47,52,51);(8,15)(11,18)(14,19)(17,20)(21,25,27,26)(22,30,32,31)(23,35,37,36)(24,40,41,39)(28,43,44,33)(29,38,46,45)(34,49,50,48)(42,51,52,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=fi,c^2=f,d^2=i,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, 6473495);