#9203687

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

4427 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=h,c^2=gh,d^2=gh,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9110883);