#9726527

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

4165 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)(3,4)(5,8)(6,7)(9,10)(11,21)(12,17)(13,18)(14,19)(15,20)(16,22)(23,24)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,38)(37,40);(1,3,5,7)(2,4,8,6)(9,11,15,22)(10,16,20,21)(12,13,19,23)(14,18,17,24)(25,27)(26,30)(28,36)(29,35)(31,39)(32,33)(34,37)(38,40);(1,3)(2,6)(4,8)(5,7)(25,26,29,32)(27,30,35,33)(28,31,37,40)(34,38,36,39);(1,2,5,8)(3,4,7,6)(9,12,15,19)(10,17,20,14)(11,13,22,23)(16,24,21,18)(25,26)(27,30)(28,31)(29,32)(33,35)(34,38)(36,39)(37,40);(1,3)(2,4)(5,7)(6,8)(9,13)(10,18)(11,19)(12,22)(14,21)(15,23)(16,17)(20,24)(25,28,29,37)(26,31,32,40)(27,34,35,36)(30,38,33,39);(1,4)(2,7)(3,8)(5,6)(9,14,15,17)(10,19,20,12)(11,18,22,24)(13,16,23,21)(25,27)(26,30)(28,36)(29,35)(31,39)(32,33)(34,37)(38,40)

Polycyclic presentation:

2^9|b^2=gh,c^2=hi,d^2=gh,e^2=hi,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9633723);