#9241070

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

4564 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9148266);