#6684435

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682360
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1739 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)(7,14)(8,12)(9,18)(10,13)(11,17)(15,20)(16,19)(21,22)(23,27)(24,35)(25,29)(26,31)(28,32)(30,34)(33,41)(36,48)(37,40)(38,44)(39,45)(42,46)(43,47)(49,52)(50,51);(1,2)(3,4)(5,7)(6,11)(8,15)(9,16)(10,17)(12,19)(13,14)(18,20)(21,23,26,34)(22,27,31,30)(24,32,37,47)(25,33,39,38)(28,40,43,35)(29,41,45,44)(36,46,50,49)(42,51,52,48);(2,4)(5,8,10,9)(6,12,13,18)(7,15,17,16)(11,19,14,20)(21,24,34,47,26,37,23,32)(22,28,27,40,31,43,30,35)(25,36,38,49,39,50,33,46)(29,42,41,51,45,52,44,48);(5,9,10,8)(6,12,13,18)(7,16,17,15)(11,19,14,20)(21,25,26,39)(22,29,31,45)(23,33,34,38)(24,36,37,50)(27,41,30,44)(28,42,43,52)(32,46,47,49)(35,48,40,51);(2,4)(7,17)(11,14)(15,16)(19,20)(21,23,26,34)(22,27,31,30)(24,32,37,47)(25,38,39,33)(28,40,43,35)(29,44,45,41)(36,49,50,46)(42,48,52,51)

Polycyclic presentation:

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

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, 6591631);