#9161748

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

4408 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=h,d^2=h,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9068944);