Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #27987 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2504 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14020 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1200 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)(3,11,13,7)(4,15)(5,10)(6,9)(8,17)(12,27,29,21)(14,28,30,22)(16,32,31,23)(18,26)(19,25)(20,24);(1,3)(2,7)(4,16,17,31)(5,14)(6,13)(8,23,15,32)(9,11)(10,22)(12,18,29,19)(20,30)(21,25,27,26)(24,28);(1,4,5,19,6,17,20,18)(2,8,24,25,9,15,10,26)(3,12,14,31,13,29,30,16)(7,21,28,23,11,27,22,32);(1,3)(2,7)(4,12)(5,14)(6,13)(8,21)(9,11)(10,22)(15,27)(16,18)(17,29)(19,31)(20,30)(23,25)(24,28)(26,32)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=eg,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,f^b=fi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72317787, #72317872, #72318474, … |
| maximal normal subgroup of | #72317787, #72317872, #72318474, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 275751);