Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454375 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1062 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,4)(6,7)(9,10)(11,13)(12,14)(15,16)(18,22)(19,25)(24,31)(27,29);(5,6,8,7)(9,11,12,15)(10,13,14,16)(17,18,21,24,20,22,28,31)(19,26,27,30,25,32,29,23);(1,2,3,4)(5,6)(7,8)(17,19,20,25)(18,23,22,30)(21,29,28,27)(24,32,31,26);(5,7,8,6)(10,14)(13,16)(19,25)(23,30)(26,32)(27,29)Polycyclic presentation:
2^9|b^2=eghi,c^2=efi,d^2=ei,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 379642);