Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33517 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C22 | C22×#107 |
| prime radical | C24 | #107 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
480 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(18,22)(19,23)(20,26)(24,28)(25,32)(27,31)(29,30);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,18,23,30,20,25,27,28,21,24,31,32,26,29,19,22);(3,9)(6,12)(10,15)(13,16);(18,24)(22,28)(25,29)(30,32)Polycyclic presentation:
2^9|a^2=f,b^2=eg,e^2=gi,f^2=h,g^2=i,b^a=be,e^a=eg,g^a=gi,c^b=ch,d^b=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72321682, #72321889, #72321901 |
| maximal normal subgroup of | #72321682, #72321889, #72321901 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390629);