#6568090

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

1353 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)(27,32)(28,39)(29,34)(30,43)(31,35)(33,41)(36,51)(37,46)(38,48)(40,50)(42,47)(44,49)(45,54)(52,56)(53,55);(1,3,5,4)(2,6,8,7)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27,31,38)(26,32,35,48)(28,36,40,52)(29,42,30,37)(33,45,49,55)(34,47,43,46)(39,51,50,56)(41,54,44,53);(1,3,5,4)(2,7,8,6)(11,17)(14,20)(18,23)(21,24)(25,28)(26,33)(27,36)(29,41)(30,44)(31,40)(32,45)(34,50)(35,49)(37,53)(38,52)(39,43)(42,54)(46,51)(47,56)(48,55);(1,4,5,3)(2,6,8,7)(9,12)(10,15)(11,18)(13,19)(14,21)(16,22)(17,23)(20,24)(28,40)(33,49)(36,52)(39,50)(41,44)(45,55)(51,56)(53,54);(12,19)(15,22)(18,23)(21,24)(25,29,31,30)(26,34,35,43)(27,37,38,42)(28,41,40,44)(32,46,48,47)(33,50,49,39)(36,53,52,54)(45,51,55,56)

Polycyclic presentation:

2^9|a^2=fh,b^2=fi,c^2=f,d^2=f,e^2=i,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, 6475286);