#6684322

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667556
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×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1751 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)(21,22,26,30)(23,27,34,43)(24,28,36,45)(25,29,39,48)(31,40,49,51)(32,41,33,42)(35,44,50,52)(37,46,38,47);(5,7)(6,11)(8,15)(9,16)(10,17)(12,19)(13,14)(18,20)(21,23,26,34)(22,27,30,43)(24,35,36,50)(25,32,39,33)(28,44,45,52)(29,41,48,42)(31,38,49,37)(40,47,51,46);(1,2,3,4)(5,8,10,9)(6,12,13,18)(7,15,17,16)(11,19,14,20)(21,24)(22,28)(23,31)(25,38)(26,36)(27,40)(29,47)(30,45)(32,35)(33,50)(34,49)(37,39)(41,44)(42,52)(43,51)(46,48);(5,9,10,8)(6,12,13,18)(7,16,17,15)(11,19,14,20)(24,36)(28,45)(31,49)(35,50)(37,38)(40,51)(44,52)(46,47);(7,17)(11,14)(15,16)(19,20)(23,32)(25,39)(27,41)(29,48)(31,35)(33,34)(37,38)(40,44)(42,43)(46,47)(49,50)(51,52)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6591518);