#6912159

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1402 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)(25,26,30,34)(27,35,38,49)(28,32,40,46)(29,42,36,33)(31,43,45,39)(37,48,51,56)(41,53,50,47)(44,54,55,52);(1,2)(3,7)(4,6)(5,8)(10,14)(12,19)(15,20)(18,23)(28,40)(32,46)(37,51)(41,50)(44,55)(47,53)(48,56)(52,54);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,27,36,43,30,38,29,39)(26,31,42,49,34,45,33,35)(28,37,50,54,40,51,41,52)(32,44,53,56,46,55,47,48);(3,8)(5,7)(10,14)(11,17)(16,21)(18,23)(25,28,30,40)(26,32,34,46)(27,37,38,51)(29,41,36,50)(31,44,45,55)(33,47,42,53)(35,48,49,56)(39,52,43,54);(2,6)(5,7)(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(27,38)(28,40)(31,45)(32,46)(35,49)(39,43)(41,50)(47,53)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6819355);