#6937398

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884841
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

1298 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=h,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6844594);