Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #187756 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#540 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2644 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)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(17,18)(19,26,28,23)(20,22)(21,24)(25,29)(27,31,32,30);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(18,22)(20,29)(23,30)(24,25)(26,31)(27,32);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,19,21,28)(18,23,24,26)(20,27,29,32)(22,30,25,31);(2,8)(6,9)(7,14)(13,15)(18,24)(22,25)(23,26)(30,31)Polycyclic presentation:
2^9|a^2=g,c^2=h,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,e^b=ei,g^b=ghi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72292871, C2×#187758, #72293038, … |
| maximal normal subgroup of | #72292871, C2×#187758, #72293038, … |
| maximal quotient of | C2×#187758, #72293045, #72293067 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 94954);