#6566273

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

1701 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)(28,36)(29,40)(34,42)(35,44)(37,43)(39,46)(41,45)(47,50)(48,52)(49,51);(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23,25,36,27,35,26,37)(22,28,31,33,32,43,30,44)(24,34,39,49,41,48,40,50)(29,42,45,47,46,52,38,51);(2,4)(7,13)(10,16)(14,19)(17,20)(21,24,25,39,27,41,26,40)(22,29,31,45,32,46,30,38)(23,34,36,49,35,48,37,50)(28,42,33,47,43,52,44,51);(5,8)(6,11)(7,14)(9,15)(10,17)(12,18)(13,19)(16,20)(23,35)(28,43)(33,44)(34,48)(36,37)(42,52)(47,51)(49,50);(8,15)(11,18)(14,19)(17,20)(21,25,27,26)(22,30,32,31)(23,36,35,37)(24,39,41,40)(28,44,43,33)(29,38,46,45)(34,49,48,50)(42,51,52,47)

Polycyclic presentation:

2^9|a^2=h,b^2=f,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^b=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, 6473469);