#6675127

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667980
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1778 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi

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, 6582323);