#8464139

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

4518 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)(10,14)(11,17)(16,22)(18,19)(25,26)(27,36)(28,32)(29,33)(30,34)(31,35)(37,38)(39,40);(3,7)(6,8)(9,10)(11,15)(12,16)(13,14)(17,21)(18,23)(19,24)(20,22)(27,28)(32,36)(37,39)(38,40);(1,2)(3,6)(4,5)(7,8)(25,27,31,28)(26,32,35,36)(29,37,34,39)(30,38,33,40);(1,3,4,7)(2,6,5,8)(9,11,13,17)(10,15,14,21)(12,19,20,18)(16,24,22,23)(25,28,31,27)(26,32,35,36)(29,39,34,37)(30,38,33,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,21)(18,23)(19,24)(20,22)(25,29,31,34)(26,33,35,30)(27,37,28,39)(32,40,36,38);(9,12,13,20)(10,16,14,22)(11,18,17,19)(15,23,21,24)(25,30,31,33)(26,34,35,29)(27,38,28,40)(32,39,36,37)

Polycyclic presentation:

2^9|c^2=i,d^2=gh,e^2=i,f^2=gi,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, 8371335);