#6568105

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1355 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=fi,c^2=f,d^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ei,d^c=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, 6475301);