#6686622

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1673 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,d^2=f,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6593818);