Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29902 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2622 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14262 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1809 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)(3,11,15,7)(4,8)(5,19)(6,9)(10,21)(12,14,23,20)(13,25,28,22)(16,17)(18,30)(24,31)(26,29,27,32);(1,3)(2,7)(4,13)(5,20)(6,15)(8,22)(9,11)(10,12)(14,21)(16,25)(17,28)(18,32)(19,23)(24,26)(27,30)(29,31);(1,2)(3,12)(4,16,17,8)(5,19)(6,9)(7,20)(10,21)(11,14)(13,27,28,26)(15,23)(18,24,31,30)(22,32,25,29);(1,4)(2,8)(3,13)(5,18)(6,17)(7,22)(9,16)(10,24)(11,25)(12,26)(14,29)(15,28)(19,30)(20,32)(21,31)(23,27)Polycyclic presentation:
2^9|a^2=h,c^2=f,g^2=i,b^a=be,c^a=cf,e^a=ei,g^a=gi,c^b=cg,g^b=gi,h^b=hi,d^c=de,e^c=ei,g^c=gi,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72338484, #72338489, #72338495 |
| maximal normal subgroup of | #72338484, #72338489, #72338495 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 287466);