#6541147

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1656 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 52 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6448343);