#8723003

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

4547 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:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8630199);