#8700325

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

4418 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=gi,d^2=h,e^2=h,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, 8607521);