Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476291 |
| 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 | |
1301 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,11,6,10,21,8)(3,12,14,23,15,7,20,25)(4,16,18,28,19,9,22,26)(13,29,17,27,30,24,31,32);(1,3)(2,7)(4,17)(5,20)(6,15)(8,25)(9,27)(10,12)(11,23)(13,18)(14,21)(16,32)(19,31)(22,30)(24,28)(26,29);(2,8)(5,21)(7,23)(9,28)(10,11)(12,25)(14,20)(16,26)(17,31)(18,22)(24,27)(29,32);(1,4)(2,9)(3,13)(5,22)(6,19)(7,24)(8,26)(10,16)(11,28)(12,29)(14,31)(15,30)(17,20)(18,21)(23,32)(25,27)Polycyclic presentation:
2^9|a^2=f,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=def,f^b=fi,h^b=hi,d^c=de,f^c=fi,f^d=fi,g^d=gi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72237054, #72237068, #72237071, … |
| maximal normal subgroup of | #72237054, #72237068, #72237071, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 383487);