#6547168

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1649 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)(21,22,27,31,26,32,25,30)(23,28,34,42,36,44,35,43)(24,37,38,29,40,45,39,46)(33,47,49,41,50,52,48,51);(1,2,3,4)(5,6)(7,11)(8,12)(9,10)(13,19)(14,17)(15,20)(16,18)(21,23,26,36)(22,28,32,44)(24,33,40,50)(25,35,27,34)(29,41,46,51)(30,43,31,42)(37,47,45,52)(38,49,39,48);(6,10)(11,17)(12,18)(19,20)(21,24,26,40)(22,29,32,46)(23,33,36,50)(25,38,27,39)(28,41,44,51)(30,45,31,37)(34,48,35,49)(42,47,43,52);(5,7,9,14)(6,11,10,17)(8,15,16,13)(12,20,18,19)(21,25,26,27)(22,30,32,31)(23,34,36,35)(24,38,40,39)(28,42,44,43)(29,45,46,37)(33,48,50,49)(41,47,51,52);(5,8,9,16)(6,12,10,18)(7,13,14,15)(11,19,17,20)(21,25,26,27)(22,31,32,30)(23,35,36,34)(24,38,40,39)(28,42,44,43)(29,37,46,45)(33,49,50,48)(41,47,51,52)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6454364);