Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #188062 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#560 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2832 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,10,5,7,15,20)(3,9,13,24,12,19,26,30)(6,8,16,22,17,18,28,29)(11,23,25,32,14,21,27,31);(2,7)(3,11)(4,15)(8,18)(9,21)(12,14)(13,27)(16,28)(19,23)(24,32)(25,26)(30,31);(2,8)(3,12)(4,16)(7,18)(9,23)(11,14)(13,25)(15,28)(19,21)(24,30)(26,27)(31,32);(1,3)(2,9)(4,13)(5,12)(6,14)(7,19)(8,21)(10,24)(11,17)(15,26)(16,27)(18,23)(20,30)(22,31)(25,28)(29,32)Polycyclic presentation:
2^9|a^2=g,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dhi,g^b=gh,d^c=dh,g^c=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72234500, #72234501, #72234502, … |
| maximal normal subgroup of | #72234500, #72234501, #72234502, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 95359);