#6687707

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
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	#30704	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1750 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fg,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=di,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, 6594903);