#6526954

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500434
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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 ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1999 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fi,c^2=g,d^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=eh,d^c=dhi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6434150);