#6568724

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535655
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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 ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1358 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6475920);