#6675229

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1781 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)(23,28,36,44)(24,30,39,48)(25,31,26,29)(33,42,37,46)(34,43,35,41)(38,45,40,47)(49,51,50,52);(1,2)(3,4)(5,7,10,17)(6,11,13,15)(8,16,9,18)(12,19,14,20)(21,23,27,36)(22,28,32,44)(24,37,39,33)(25,34,26,35)(29,41,31,43)(30,46,48,42)(38,50,40,49)(45,52,47,51);(2,4)(5,8,10,9)(6,12,13,14)(7,16,17,18)(11,19,15,20)(22,29)(25,26)(28,41)(30,45)(31,32)(34,35)(38,40)(42,51)(43,44)(46,52)(47,48)(49,50);(6,13)(11,15)(12,14)(19,20)(21,24)(22,30)(23,33)(25,40)(26,38)(27,39)(28,42)(29,45)(31,47)(32,48)(34,50)(35,49)(36,37)(41,51)(43,52)(44,46);(2,4)(5,9,10,8)(6,14,13,12)(7,16,17,18)(11,19,15,20)(21,25,27,26)(22,31,32,29)(23,34,36,35)(24,38,39,40)(28,43,44,41)(30,45,48,47)(33,49,37,50)(42,51,46,52)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,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,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6582425);