#6684331

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×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1737 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)(21,22,27,32)(23,28,36,44)(24,29,26,31)(25,30,39,47)(33,41,34,42)(35,43,50,52)(37,45,40,48)(38,46,49,51);(5,7)(6,11)(8,15)(9,16)(10,17)(12,19)(13,14)(18,20)(21,23,27,36)(22,28,32,44)(24,34,26,33)(25,38,39,49)(29,42,31,41)(30,46,47,51)(35,40,50,37)(43,48,52,45);(1,2,3,4)(5,8,10,9)(6,12,13,18)(7,15,17,16)(11,19,14,20)(23,33)(24,26)(28,41)(29,31)(34,36)(35,38)(37,40)(42,44)(43,46)(45,48)(49,50)(51,52);(5,9,10,8)(6,12,13,18)(7,16,17,15)(11,19,14,20)(21,24,27,26)(22,29,32,31)(23,34,36,33)(25,37,39,40)(28,42,44,41)(30,45,47,48)(35,49,50,38)(43,51,52,46);(7,17)(11,14)(15,16)(19,20)(21,25,27,39)(22,30,32,47)(23,35,36,50)(24,37,26,40)(28,43,44,52)(29,45,31,48)(33,38,34,49)(41,46,42,51)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fg,d^2=fi,e^2=i,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, 6591527);