Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393889 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2615 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2710 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,4)(9,10)(11,13)(12,14)(15,16)(17,18)(19,24)(20,25)(21,30)(22,26)(23,27)(28,31)(29,32);(5,6)(7,8)(9,11,12,15)(10,13,14,16)(17,19)(18,23)(20,27)(21,29)(22,28)(24,30)(25,31)(26,32);(2,4)(6,8)(17,19)(18,24)(20,29)(21,27)(22,28)(23,30)(25,32)(26,31);(10,14)(13,16)(17,20,22,21)(18,25,26,30)(19,27,28,29)(23,31,32,24)Polycyclic presentation:
2^9|b^2=h,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 312372);