#6686451

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1673 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)(8,17)(9,12)(10,13)(11,16)(15,19)(18,20)(21,22,27,31)(23,28,26,40)(24,34,37,47)(25,30,38,44)(29,32,43,36)(33,41,39,50)(35,46,48,52)(42,45,51,49);(1,2)(3,4)(5,7)(6,11)(8,15)(9,18)(10,16)(12,19)(13,14)(17,20)(21,23,27,26)(22,28,31,40)(24,32,37,36)(25,33,38,39)(29,34,43,47)(30,41,44,50)(35,45,48,49)(42,46,51,52);(2,4)(21,24,23,32,27,37,26,36)(22,29,40,47,31,43,28,34)(25,35,33,45,38,48,39,49)(30,42,50,52,44,51,41,46);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(21,25)(22,30)(23,33)(24,35)(26,39)(27,38)(28,41)(29,42)(31,44)(32,45)(34,46)(36,49)(37,48)(40,50)(43,51)(47,52);(2,4)(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(25,38)(30,44)(33,39)(35,48)(41,50)(42,51)(45,49)(46,52)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gi,d^2=f,e^2=f,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, 6593647);