#6686519

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682532
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

1670 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)(21,22,26,30)(23,27,32,40)(24,34,36,49)(25,38,35,29)(28,42,43,37)(31,44,46,52)(33,48,45,41)(39,50,51,47);(1,2)(3,4)(5,7)(6,11)(8,15)(9,18)(10,16)(12,19)(13,14)(17,20)(21,23)(22,27)(24,31)(25,33)(26,32)(28,39)(29,41)(30,40)(34,44)(35,45)(36,46)(37,47)(38,48)(42,50)(43,51)(49,52);(2,4)(21,24,35,42,26,36,25,37)(22,28,38,49,30,43,29,34)(23,31,45,50,32,46,33,47)(27,39,48,52,40,51,41,44);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(23,32)(27,40)(31,46)(33,45)(39,51)(41,48)(44,52)(47,50);(2,4)(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(24,36)(28,43)(31,46)(34,49)(37,42)(39,51)(44,52)(47,50)

Polycyclic presentation:

2^9|a^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,d^b=di,e^b=efh,e^c=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6593715);