#9843660

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

4243 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)(11,14)(12,19)(13,16)(15,21)(17,24)(18,23)(20,22)(26,32)(28,36)(33,38)(34,37);(1,2,5,7)(3,4,8,6)(9,10)(11,14)(12,15)(13,16)(17,22)(18,23)(19,21)(20,24)(25,26)(27,33)(28,30)(29,38)(31,32)(34,35)(36,40)(37,39);(1,3)(2,6)(4,7)(5,8)(9,10,13,16)(11,14,18,23)(12,15,21,19)(17,22,20,24)(25,27,31,29)(26,33,32,38)(28,34,36,37)(30,35,40,39);(9,10)(11,14)(12,15)(13,16)(17,22)(18,23)(19,21)(20,24)(25,28)(26,30)(27,34)(29,37)(31,36)(32,40)(33,35)(38,39);(1,4,5,6)(2,3,7,8)(9,11,13,18)(10,14,16,23)(12,20,21,17)(15,24,19,22)(25,29,31,27)(26,33,32,38)(28,37,36,34)(30,35,40,39);(1,3)(2,4)(5,8)(6,7)(9,12,13,21)(10,15,16,19)(11,17,18,20)(14,22,23,24)(25,30)(26,28)(27,35)(29,39)(31,40)(32,36)(33,34)(37,38)

Polycyclic presentation:

2^9|b^2=hi,c^2=gh,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9750856);