#8436290

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

4410 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)(9,10,14,13)(11,17,19,15)(12,20,21,16)(18,22,24,23)(25,26,32,31)(27,36,38,33)(28,34,29,39)(30,40,37,35);(3,7)(6,8)(9,11,14,19)(10,15,13,17)(12,18,21,24)(16,22,20,23)(25,27)(26,33)(28,30)(29,37)(31,36)(32,38)(34,35)(39,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,18)(13,17)(14,19)(16,22)(20,23)(21,24)(25,28,32,29)(26,34,31,39)(27,30,38,37)(33,35,36,40);(1,3,4,7)(2,6,5,8)(9,11)(10,15)(12,18)(13,17)(14,19)(16,22)(20,23)(21,24)(25,29,32,28)(26,34,31,39)(27,37,38,30)(33,35,36,40);(1,2)(3,6)(4,5)(7,8)(9,12,14,21)(10,16,13,20)(11,18,19,24)(15,22,17,23)(25,30,32,37)(26,35,31,40)(27,28,38,29)(33,34,36,39);(9,13,14,10)(11,17,19,15)(12,20,21,16)(18,23,24,22)(25,31,32,26)(27,36,38,33)(28,39,29,34)(30,40,37,35)

Polycyclic presentation:

2^9|a^2=g,b^2=i,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, 8343486);