#6497383

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470922
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1314 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)(3,6)(4,5)(7,8)(9,10,13,16)(11,14,17,21)(12,19,20,15)(18,24,23,22)(25,26)(27,37)(28,42)(29,34)(30,36)(31,35)(32,40)(33,46)(38,47)(39,48)(41,49)(43,50)(44,52)(45,51)(53,54)(55,56);(2,5)(6,8)(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,27,30,39,31,41,29,40)(26,32,34,49,35,48,36,37)(28,43,46,55,45,38,44,54)(33,50,42,53,52,47,51,56);(1,3,4,7)(2,6,5,8)(11,17)(14,21)(18,23)(22,24)(25,28,29,44,31,45,30,46)(26,33,36,51,35,52,34,42)(27,38,40,55,41,43,39,54)(32,47,37,53,48,50,49,56);(11,17)(14,21)(18,23)(22,24)(25,29,31,30)(26,34,35,36)(27,39,41,40)(28,44,45,46)(32,37,48,49)(33,42,52,51)(38,54,43,55)(47,53,50,56);(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(26,35)(28,45)(32,48)(34,36)(37,49)(38,43)(44,46)(54,55)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=f,d^2=i,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^c=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, 6404579);