#8926731

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

4302 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=gi,e^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8833927);