#6675157

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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 ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1766 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)(21,22,27,32)(23,28,37,45)(24,30,39,46)(25,31,26,41)(29,40,47,38)(33,48,34,43)(35,44,36,51)(42,50,52,49);(1,2)(3,4)(5,7,10,17)(6,11,13,15)(8,16,9,18)(12,19,14,20)(21,23)(22,28)(24,33)(25,35)(26,36)(27,37)(29,42)(30,48)(31,44)(32,45)(34,39)(38,49)(40,50)(41,51)(43,46)(47,52);(2,4)(5,8,10,9)(6,12,13,14)(7,16,17,18)(11,19,15,20)(21,24)(22,29)(23,33)(25,38)(26,40)(27,39)(28,42)(30,41)(31,46)(32,47)(34,37)(35,49)(36,50)(43,44)(45,52)(48,51);(6,13)(11,15)(12,14)(19,20)(21,24)(22,30)(23,34)(25,40)(26,38)(27,39)(28,43)(29,41)(31,47)(32,46)(33,37)(35,49)(36,50)(42,44)(45,48)(51,52);(2,4)(5,9,10,8)(6,14,13,12)(7,16,17,18)(11,19,15,20)(21,25,27,26)(22,31,32,41)(23,35,37,36)(24,38,39,40)(28,44,45,51)(29,46,47,30)(33,49,34,50)(42,43,52,48)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,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,e^d=ei

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, 6582353);