#6547144

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, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1626 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 52 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=i,d^2=hi,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ei,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, 6454340);