#6567908

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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

1352 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6475104);