#9123866

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

4715 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)(11,13)(12,20)(14,17)(15,19)(16,21)(18,22)(23,24)(25,26)(27,31)(28,32)(29,40)(30,34)(33,37)(35,36)(38,39);(1,2)(3,6)(4,7)(5,8)(9,11)(10,15)(12,18)(13,17)(14,19)(16,22)(20,23)(21,24)(25,27,30,36)(26,31,34,35)(28,37,39,29)(32,33,38,40);(1,3)(2,5)(4,8)(6,7)(12,21)(16,20)(18,24)(22,23)(25,26,30,34)(27,35,36,31)(28,32,39,38)(29,33,37,40);(3,8)(5,6)(9,12)(10,16)(11,18)(13,23)(14,21)(15,22)(17,20)(19,24)(25,26)(27,31)(28,38)(29,33)(30,34)(32,39)(35,36)(37,40);(1,3)(2,6)(4,8)(5,7)(9,13,14,15)(10,11,17,19)(12,22,21,23)(16,24,20,18)(25,28,30,39)(26,32,34,38)(27,29,36,37)(31,40,35,33);(1,2)(3,5)(4,7)(6,8)(9,11)(10,13)(12,18)(14,19)(15,17)(16,23)(20,22)(21,24)(25,29)(26,33)(27,28)(30,37)(31,38)(32,35)(34,40)(36,39)

Polycyclic presentation:

2^9|b^2=h,c^2=h,e^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, 9031062);