#6548922

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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

1261 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 56 points:

(1,2,4,6)(3,7,8,5)(25,26,29,33,30,34,43,49)(27,35,38,51,37,31,44,46)(28,39,42,48,41,47,40,32)(36,52,55,45,53,56,54,50);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,16)(13,14)(17,23)(18,21)(19,24)(20,22)(25,27,30,37)(26,31,34,35)(28,36,41,53)(29,38,43,44)(32,45,48,50)(33,46,49,51)(39,56,47,52)(40,54,42,55);(10,14)(15,21)(16,22)(23,24)(25,28,30,41)(26,32,34,48)(27,36,37,53)(29,40,43,42)(31,45,35,50)(33,47,49,39)(38,54,44,55)(46,52,51,56);(9,11,13,18)(10,15,14,21)(12,19,20,17)(16,24,22,23)(27,37)(31,35)(36,53)(38,44)(45,50)(46,51)(52,56)(54,55);(9,12,13,20)(10,16,14,22)(11,17,18,19)(15,23,21,24)(25,29,30,43)(26,33,34,49)(27,38,37,44)(28,40,41,42)(31,46,35,51)(32,47,48,39)(36,54,53,55)(45,52,50,56)

Polycyclic presentation:

2^9|a^2=fgi,b^2=f,c^2=i,d^2=h,e^2=hi,g^2=i,b^a=bf,c^a=cg,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, 6456118);