Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394271 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#392 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1713 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,19)(12,16)(13,26)(14,18)(15,21)(17,28)(20,35)(22,33)(23,34)(24,38)(25,32)(27,30)(29,37)(31,39)(36,40);(1,3,4,7)(2,5,6,8)(9,11)(10,15)(12,23)(13,27)(14,22)(16,29)(17,33)(18,30)(19,26)(20,25)(21,28)(24,36)(31,35)(32,40)(34,38)(37,39);(9,12,14,25)(10,16,18,32)(11,20)(13,24,28,39)(15,29)(17,31,26,38)(19,34)(21,36)(22,23)(27,37)(30,40)(33,35);(2,6)(5,8)Polycyclic presentation:
2^9|b^2=h,c^2=fi,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 313402);