Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #359901 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2342 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13877 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2151 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,11)(3,14)(4,20,23,32)(5,24)(6,13,26,12)(8,19)(9,22,29,16)(10,15)(17,28)(18,31)(21,30)(25,27);(1,3)(2,8)(4,21)(5,16)(6,17)(7,19)(9,10)(11,14)(12,28)(13,31)(15,22)(18,26)(20,27)(23,25)(24,29)(30,32);(1,4,6,16,7,23,26,22)(2,9,12,20,11,29,13,32)(3,15,17,21,19,5,18,25)(8,27,28,24,14,30,31,10);(1,5)(2,10)(3,16)(4,18)(6,25)(7,15)(8,9)(11,24)(12,27)(13,30)(14,29)(17,23)(19,22)(20,28)(21,26)(31,32)Polycyclic presentation:
2^9|a^2=h,c^2=f,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,g^b=gi,h^b=hi,g^c=gi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72286050, #72286186, #72286253, … |
| maximal normal subgroup of | #72286050, #72286186, #72286253, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 267756);