#8270521

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4686 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

(1,2)(3,5)(4,8)(6,7)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,26)(27,32)(28,30)(29,33)(31,35)(34,39)(36,40)(37,38);(2,7)(3,4)(10,14)(15,17)(16,21)(22,23)(25,27,31,39)(26,32,35,34)(28,37,29,36)(30,38,33,40);(2,7)(5,8)(12,20)(16,21)(18,24)(22,23)(25,28,31,29)(26,33,35,30)(27,36,39,37)(32,38,34,40);(1,3,6,4)(2,8,7,5)(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,29,31,28)(26,33,35,30)(27,37,39,36)(32,38,34,40);(1,4,6,3)(2,5,7,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,27)(26,34)(28,36)(29,37)(30,38)(31,39)(32,35)(33,40);(1,5)(2,3)(4,7)(6,8)(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)(25,30,31,33)(26,28,35,29)(27,38,39,40)(32,37,34,36)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=gi,e^2=hi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8177717);