#6481318

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470428
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1301 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)(3,8)(4,6)(5,7)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(25,26,30,33)(27,35,36,48)(28,32,39,47)(29,41,42,34)(31,43,44,38)(37,49,50,56)(40,53,52,46)(45,54,55,51);(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27,29,38,30,36,42,43)(26,31,34,35,33,44,41,48)(28,37,40,51,39,50,52,54)(32,45,46,49,47,55,53,56);(1,3,5,4)(2,6,7,8)(9,12,13,20)(10,15,16,24)(11,17,18,19)(14,21,22,23)(27,36)(31,44)(35,48)(37,50)(38,43)(45,55)(49,56)(51,54);(11,18)(14,22)(17,19)(21,23)(25,28)(26,32)(27,37)(29,40)(30,39)(31,45)(33,47)(34,46)(35,49)(36,50)(38,51)(41,53)(42,52)(43,54)(44,55)(48,56);(1,4,5,3)(2,6,7,8)(10,16)(14,22)(15,24)(21,23)(26,33)(28,39)(31,44)(34,41)(35,48)(37,50)(40,52)(51,54)

Polycyclic presentation:

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

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, 6388514);