#8690968

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

4605 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=g,f^2=g,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, 8598164);