Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #384166 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2377 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14373 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1067 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,7,10,5,20)(3,12,13,22,15,8,16,24)(4,17,18,26,19,9,21,27)(14,25,30,32,29,23,28,31);(1,3)(2,8)(4,14)(5,13)(6,16)(7,15)(9,25)(10,12)(11,22)(17,23)(18,28)(19,29)(20,24)(21,30)(26,32)(27,31);(3,13,15,16)(8,22,12,24)(14,28,29,30)(23,31,25,32);(1,4)(2,9)(3,14)(5,18)(6,21)(7,19)(8,23)(10,17)(11,26)(12,25)(13,28)(15,29)(16,30)(20,27)(22,31)(24,32)Polycyclic presentation:
2^9|a^2=h,c^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72338601, #72338602, #72338610 |
| maximal normal subgroup of | #72338601, #72338602, #72338610 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 291363);