#6544090

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

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

2741 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)(21,22)(23,33)(24,38)(25,30)(26,31)(27,32)(28,36)(29,40)(34,48)(35,44)(37,43)(39,46)(41,45)(42,51)(47,50)(49,52);(2,4)(5,6)(7,11)(8,12)(9,10)(13,19)(14,17)(15,20)(16,18)(21,23,27,37)(22,28,32,44)(24,34,41,51)(25,36,26,35)(29,42,46,48)(30,43,31,33)(38,52,45,47)(39,50,40,49);(6,10)(11,17)(12,18)(19,20)(21,24,27,41)(22,29,32,46)(23,34,37,51)(25,39,26,40)(28,42,44,48)(30,38,31,45)(33,47,43,52)(35,49,36,50);(5,7,9,14)(6,11,10,17)(8,15,16,13)(12,20,18,19)(21,25,27,26)(22,30,32,31)(23,35,37,36)(24,39,41,40)(28,33,44,43)(29,38,46,45)(34,49,51,50)(42,47,48,52);(5,8,9,16)(6,12,10,18)(7,13,14,15)(11,19,17,20)(21,25,27,26)(22,31,32,30)(23,36,37,35)(24,39,41,40)(28,33,44,43)(29,45,46,38)(34,50,51,49)(42,47,48,52)

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=i,d^2=hi,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,g^a=gi,c^b=ch,d^b=di,f^b=fi,g^b=gi,f^c=fi,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, 6451286);