Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418851 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2679 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15266 |
| prime radical | #60398 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1934 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,17,7,4,15,10)(3,11,13,22,14,8,20,25)(5,9,18,29,19,16,28,24)(12,21,26,32,27,23,30,31);(1,3)(2,8)(4,11)(5,12)(6,20)(7,14)(9,23)(10,25)(13,15)(16,21)(17,22)(18,30)(19,27)(24,31)(26,28)(29,32);(1,4)(2,7)(3,8)(5,16)(6,10)(9,19)(11,14)(12,21)(13,25)(15,17)(18,24)(20,22)(23,27)(26,32)(28,29)(30,31);(1,5)(2,9)(3,12)(4,16)(6,18)(7,19)(8,21)(10,24)(11,23)(13,26)(14,27)(15,28)(17,29)(20,30)(22,31)(25,32)Polycyclic presentation:
2^9|a^2=ef,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,f^b=fi,h^b=hi,e^c=ei,f^c=fi,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72318864, #72318890, #72318925, … |
| maximal normal subgroup of | #72318864, #72318890, #72318925, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326047);