Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32932 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1534 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)(5,6,7,8)(9,10)(11,13)(12,14)(15,16)(18,23)(19,21)(20,28)(24,26)(25,29)(27,30);(9,11,12,15)(10,13,14,16)(17,18,19,24,22,26,21,23)(20,29,27,31,30,25,28,32);(10,14)(13,16)(17,19,22,21)(18,24,26,23)(20,28,30,27)(25,31,29,32);(2,4)(17,20,21,28,22,30,19,27)(18,25,23,31,26,29,24,32)Polycyclic presentation:
2^9|a^2=h,b^2=ef,c^2=i,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,g^a=gi,d^b=di,d^c=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72308314 |
| maximal normal subgroup of | #72308314 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299612);