Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #457608 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2735 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #16391 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1437 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,12,8,11,6,21)(3,13,17,28,18,9,16,27)(4,19,10,24,22,25,23,5)(14,31,20,26,30,15,32,29);(1,3)(2,9)(4,20)(5,15)(6,17)(7,16)(8,18)(10,14)(11,13)(12,28)(19,26)(21,27)(22,32)(23,30)(24,31)(25,29);(1,4,6,23,8,22,7,10)(2,5,12,19,11,24,21,25)(3,14,16,32,18,30,17,20)(9,26,27,15,13,29,28,31);(1,5)(2,10)(3,15)(4,21)(6,25)(7,19)(8,24)(9,14)(11,23)(12,22)(13,30)(16,26)(17,29)(18,31)(20,27)(28,32)Polycyclic presentation:
2^9|a^2=g,c^2=f,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,f^b=fi,g^b=gi,d^c=de,e^c=ei,f^d=fi,h^d=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72318583, #72318876, #72318890, … |
| maximal normal subgroup of | #72318583, #72318876, #72318890, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 368627);