#9160928

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

4521 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,e^2=i,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, 9068124);