#6568228

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

1351 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)(27,36)(28,40)(29,34)(30,35)(31,45)(32,38)(33,43)(37,52)(39,48)(41,49)(42,50)(44,47)(46,55)(51,54)(53,56);(1,3,5,4)(2,6,8,7)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27,30,38)(26,32,35,36)(28,37,42,53)(29,44,31,39)(33,46,49,51)(34,48,45,47)(40,56,50,52)(41,55,43,54);(1,3,5,4)(2,7,8,6)(11,17)(14,20)(18,23)(21,24)(25,28,30,42)(26,33,35,49)(27,37,38,53)(29,43,31,41)(32,46,36,51)(34,50,45,40)(39,55,44,54)(47,56,48,52);(1,4,5,3)(2,6,8,7)(9,12)(10,15)(11,18)(13,19)(14,21)(16,22)(17,23)(20,24)(27,38)(32,36)(37,53)(39,44)(46,51)(47,48)(52,56)(54,55);(12,19)(15,22)(18,23)(21,24)(25,29,30,31)(26,34,35,45)(27,39,38,44)(28,41,42,43)(32,47,36,48)(33,40,49,50)(37,54,53,55)(46,52,51,56)

Polycyclic presentation:

2^9|a^2=fhi,b^2=f,c^2=f,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ei,e^c=ei,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, 6475424);