#6684339

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667604
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

1704 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)(8,12)(9,18)(10,13)(11,17)(15,20)(16,19)(21,22,26,29)(23,32,36,28)(24,37,27,30)(25,39,41,31)(33,43,34,44)(35,45,51,49)(38,46,42,48)(40,47,50,52);(5,7)(6,11)(8,15)(9,16)(10,17)(12,19)(13,14)(18,20)(21,23,26,36)(22,28,29,32)(24,34,27,33)(25,40,41,50)(30,44,37,43)(31,47,39,52)(35,38,51,42)(45,48,49,46);(1,2,3,4)(5,8,10,9)(6,12,13,18)(7,15,17,16)(11,19,14,20)(22,29)(23,33)(24,27)(28,43)(31,39)(32,44)(34,36)(35,50)(38,42)(40,51)(45,47)(49,52);(5,9,10,8)(6,12,13,18)(7,16,17,15)(11,19,14,20)(21,24,26,27)(22,30,29,37)(23,34,36,33)(25,38,41,42)(28,44,32,43)(31,46,39,48)(35,40,51,50)(45,52,49,47);(7,17)(11,14)(15,16)(19,20)(21,25,26,41)(22,31,29,39)(23,35,36,51)(24,38,27,42)(28,45,32,49)(30,46,37,48)(33,50,34,40)(43,47,44,52)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6591535);