#8436291

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

4414 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)(11,13)(12,20)(14,15)(16,18)(17,22)(19,23)(21,24)(25,26)(27,35)(28,34)(29,31)(30,33)(32,36)(37,39)(38,40);(3,7)(6,8)(10,15)(11,18)(17,20)(19,24)(25,27,32,30)(26,33,36,35)(28,37,29,38)(31,39,34,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,19)(13,15)(14,18)(17,23)(20,21)(22,24)(25,28,32,29)(26,34,36,31)(27,37,30,38)(33,40,35,39);(1,3,4,7)(2,6,5,8)(9,11)(10,16)(12,19)(13,15)(14,18)(17,23)(20,21)(22,24)(25,29,32,28)(26,34,36,31)(27,38,30,37)(33,40,35,39);(1,2)(3,6)(4,5)(7,8)(9,12,14,22)(10,17,15,20)(11,19,18,24)(13,21,16,23)(25,30,32,27)(26,35,36,33)(28,38,29,37)(31,40,34,39);(9,13,14,16)(10,11,15,18)(12,21,22,23)(17,19,20,24)(25,31,32,34)(26,29,36,28)(27,39,30,40)(33,38,35,37)

Polycyclic presentation:

2^9|b^2=gi,c^2=gi,d^2=gh,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, 8343487);