#8733521

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

4497 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8640717);