#8508886

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

4247 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)(3,7)(10,14)(11,17)(16,22)(18,19)(25,26,29,32)(27,33,35,30)(28,31,36,38)(34,39,40,37);(1,2,4,5)(3,6,7,8)(9,10,13,14)(11,15,17,21)(12,16,20,22)(18,23,19,24)(28,36)(31,38)(34,40)(37,39);(1,2)(3,6)(4,5)(7,8)(25,27,29,35)(26,30,32,33)(28,34,36,40)(31,37,38,39);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,19)(13,17)(14,21)(16,24)(18,20)(22,23)(25,28,29,36)(26,31,32,38)(27,34,35,40)(30,37,33,39);(1,3,4,7)(2,6,5,8)(9,10)(11,15)(12,16)(13,14)(17,21)(18,23)(19,24)(20,22)(25,26)(27,30)(28,31)(29,32)(33,35)(34,37)(36,38)(39,40);(9,12,13,20)(10,16,14,22)(11,18,17,19)(15,23,21,24)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8416082);