#6686001

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1714 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)(21,22,26,30)(23,31,35,47)(24,28,37,44)(25,39,32,29)(27,40,42,34)(33,46,36,50)(38,52,48,45)(41,51,43,49);(1,2)(3,4)(5,7,9,15)(6,10,11,13)(8,16,17,14)(12,20,19,18);(2,4)(21,23,32,40,26,35,25,34)(22,27,39,47,30,42,29,31)(24,36,48,49,37,33,38,51)(28,43,52,46,44,41,45,50);(6,11)(10,13)(12,19)(18,20)(21,24,26,37)(22,28,30,44)(23,33,35,36)(25,38,32,48)(27,41,42,43)(29,45,39,52)(31,46,47,50)(34,49,40,51);(2,4)(5,8,9,17)(6,12,11,19)(7,14,15,16)(10,18,13,20)(24,37)(28,44)(33,36)(38,48)(41,43)(45,52)(46,50)(49,51)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=gi,d^2=i,e^2=f,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, 6593197);