Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #180873 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C22×#95 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1401 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,9,11,15)(4,7)(5,8)(6,10,13,16)(12,14)(17,18,20,23,21,24,27,30)(19,22,26,29,25,28,31,32);(1,3)(2,6,8,13)(4,10)(5,11)(7,9,14,15)(12,16);(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32);(19,25)(22,28)(26,31)(29,32)Polycyclic presentation:
2^9|a^2=f,b^2=g,e^2=h,f^2=i,b^a=be,g^a=gh,f^b=fh,d^c=di110 conjugacy classes — view the full character table.
| maximal subgroup of | #72279567, #72279890, #72280217, … |
| maximal normal subgroup of | #72279567, #72279890, #72280217, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 88069);