#6564732

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1780 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6471928);