#8700323

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

4472 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=ghi,d^2=g,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=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, 8607519);