#6566309

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535664
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

1710 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)(29,42)(34,43)(36,45)(37,44)(39,46)(40,48)(41,47)(49,51)(50,52);(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23,26,36,27,37,25,35)(22,28,30,44,32,45,31,33)(24,39,42,50,40,34,41,51)(29,46,38,49,47,43,48,52);(2,4)(7,13)(10,16)(14,19)(17,20)(21,24,25,41,27,40,26,42)(22,29,31,48,32,47,30,38)(23,34,35,50,37,39,36,51)(28,43,33,49,45,46,44,52);(5,8)(6,11)(7,14)(9,15)(10,17)(12,18)(13,19)(16,20)(24,40)(29,47)(34,39)(38,48)(41,42)(43,46)(49,52)(50,51);(8,15)(11,18)(14,19)(17,20)(21,25,27,26)(22,30,32,31)(23,35,37,36)(24,41,40,42)(28,44,45,33)(29,38,47,48)(34,50,39,51)(43,52,46,49)

Polycyclic presentation:

2^9|a^2=hi,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=ehi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6473505);