Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419041 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2424 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#454 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1861 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(6,12)(7,17)(9,10)(13,18)(14,22)(19,33)(20,21)(23,24)(26,28)(27,36)(31,34)(32,35);(1,2)(3,4)(5,6,9,15,11,12,10,16)(7,18,17,13,20,28,21,26)(8,22,23,30,25,14,24,29)(19,32,33,35,34,36,31,27);(2,4)(5,7)(6,13)(8,19)(9,21)(10,17)(11,20)(12,26)(14,27)(15,18)(16,28)(22,35)(23,31)(24,33)(25,34)(29,36)(30,32);(5,8,11,25)(6,14,12,22)(7,19,20,34)(9,24,10,23)(13,27,26,35)(15,30,16,29)(17,31,21,33)(18,32,28,36)Polycyclic presentation:
2^9|b^2=f,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^d=fi,g^d=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 328053);