#8431683

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

4583 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:

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

Polycyclic presentation:

2^9|b^2=g,d^2=g,e^2=g,f^2=gh,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, 8338879);