#6568786

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535664
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	C₂×#555	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1383 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)(27,36)(28,41)(29,34)(30,46)(31,35)(32,38)(33,44)(37,50)(39,49)(40,48)(42,47)(43,51)(45,56)(52,53)(54,55);(2,5)(6,8)(9,11,13,17)(10,14,15,21)(12,18,19,23)(16,20,22,24)(25,27,29,39,31,40,30,38)(26,32,46,48,35,49,34,36)(28,42,43,53,45,37,44,54)(33,50,56,52,51,47,41,55);(1,3,4,7)(2,6,5,8)(10,15)(11,17)(16,22)(18,23)(25,28,29,43,31,45,30,44)(26,33,46,56,35,51,34,41)(27,37,39,54,40,42,38,53)(32,47,48,55,49,50,36,52);(9,12)(10,16)(11,18)(13,19)(14,20)(15,22)(17,23)(21,24)(25,29,31,30)(26,34,35,46)(27,38,40,39)(28,43,45,44)(32,48,49,36)(33,41,51,56)(37,53,42,54)(47,55,50,52);(12,19)(16,22)(18,23)(20,24)(25,29,31,30)(26,34,35,46)(27,39,40,38)(28,43,45,44)(32,36,49,48)(33,41,51,56)(37,54,42,53)(47,52,50,55)

Polycyclic presentation:

2^9|a^2=fgh,b^2=gi,c^2=fi,d^2=i,e^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^d=ehi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6475982);