Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455397 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2700 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #16299 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
860 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)(3,8)(4,16,18,27)(5,20)(6,10)(7,11)(9,19,26,21)(12,28)(13,30,17,32)(14,24)(15,22)(23,29,25,31);(3,12,14,15)(8,22,24,28)(13,29,17,31)(23,32,25,30);(1,3)(2,8)(4,17,18,13)(5,15)(6,12)(7,14)(9,25,26,23)(10,28)(11,24)(16,30,27,32)(19,29,21,31)(20,22);(1,4)(2,9)(3,13)(5,19)(6,21)(7,18)(8,23)(10,27)(11,26)(12,29)(14,17)(15,31)(16,20)(22,32)(24,25)(28,30)Polycyclic presentation:
2^9|a^2=f,b^2=e,c^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^c=de,e^c=ei,g^c=gi,h^c=hi,f^d=fi,h^d=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72319622, #72320009, #72320054, … |
| maximal normal subgroup of | #72319622, #72320009, #72320054, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 364875);