#8712769

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

4540 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=gh,d^2=gh,e^2=gh,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, 8619965);