#8433736

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

4687 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8340932);