Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476289 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17086 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1295 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,11,7,8,6,12)(3,13,15,22,16,27,21,9)(4,17,18,10,19,25,20,23)(14,24,29,32,30,26,31,28);(2,8)(4,18)(5,6)(10,25)(13,27)(14,29)(15,21)(17,23)(19,20)(24,28)(26,32)(30,31);(1,3)(2,9)(4,14)(5,21)(6,15)(7,16)(8,22)(10,26)(11,27)(12,13)(17,28)(18,31)(19,30)(20,29)(23,24)(25,32);(1,4)(2,10)(3,14)(5,20)(6,18)(7,19)(8,23)(9,24)(11,17)(12,25)(13,28)(15,31)(16,30)(21,29)(22,26)(27,32)Polycyclic presentation:
2^9|a^2=f,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=def,f^b=fi,h^b=hi,d^c=de,f^c=fi,f^d=fi,g^d=gi,h^d=hi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72237054, #72237063, #72237072, … |
| maximal normal subgroup of | #72237054, #72237063, #72237072, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 383485);