#6686085

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1675 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)(7,13)(8,12)(9,11)(10,15)(14,20)(16,18)(17,19)(21,22,27,31)(23,32,35,43)(24,29,36,45)(25,30,26,41)(28,39,44,34)(33,49,37,52)(38,47,40,48)(42,50,46,51);(1,2)(3,4)(5,7,9,15)(6,10,11,13)(8,16,17,14)(12,20,19,18)(24,36)(29,45)(33,37)(38,40)(42,46)(47,48)(49,52)(50,51);(2,4)(21,23,25,39,27,35,26,34)(22,28,41,43,31,44,30,32)(24,37,40,51,36,33,38,50)(29,46,47,49,45,42,48,52);(6,11)(10,13)(12,19)(18,20)(21,24)(22,29)(23,33)(25,40)(26,38)(27,36)(28,42)(30,48)(31,45)(32,49)(34,51)(35,37)(39,50)(41,47)(43,52)(44,46);(2,4)(5,8,9,17)(6,12,11,19)(7,14,15,16)(10,18,13,20)(21,25,27,26)(22,30,31,41)(23,34,35,39)(24,38,36,40)(28,43,44,32)(29,47,45,48)(33,50,37,51)(42,49,46,52)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=gi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di,e^c=ei,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, 6593281);