#6674582

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×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1801 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)(21,22,27,31)(23,28,26,39)(24,34,36,41)(25,30,38,46)(29,40,44,32)(33,42,37,45)(35,48,50,51)(43,49,52,47);(1,2)(3,4)(5,7)(6,11)(8,16)(9,18)(10,17)(12,19)(13,15)(14,20)(21,23,27,26)(22,28,31,39)(24,32,36,40)(25,37,38,33)(29,41,44,34)(30,45,46,42)(35,49,50,47)(43,48,52,51);(2,4)(5,8,10,9)(6,12,13,14)(7,16,17,18)(11,19,15,20)(21,24)(22,29)(23,32)(25,35)(26,40)(27,36)(28,41)(30,43)(31,44)(33,47)(34,39)(37,49)(38,50)(42,51)(45,48)(46,52);(6,13)(11,15)(12,14)(19,20)(21,25,27,38)(22,30,31,46)(23,33,26,37)(24,35,36,50)(28,42,39,45)(29,43,44,52)(32,47,40,49)(34,48,41,51);(2,4)(5,9,10,8)(6,14,13,12)(7,16,17,18)(11,19,15,20)(24,36)(25,38)(29,44)(30,46)(32,40)(33,37)(34,41)(42,45)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=f,d^2=i,e^2=f,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, 6581778);