Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #384777 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2337 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14389 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
827 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,9,6,10,22,11)(3,12,15,24,16,25,14,7)(4,17,20,27,21,8,19,28)(13,29,30,23,31,26,18,32);(1,3)(2,7)(4,18)(5,14)(6,16)(8,26)(9,25)(10,24)(11,12)(13,19)(15,22)(17,29)(20,31)(21,30)(23,28)(27,32);(1,4,5,19,6,21,22,20)(2,8,9,27,10,17,11,28)(3,13,15,18,16,31,14,30)(7,23,12,29,24,32,25,26);(2,9,10,11)(3,14,16,15)(7,24)(8,27,17,28)(12,25)(13,30,31,18)(23,32)(26,29)Polycyclic presentation:
2^9|a^2=h,c^2=e,d^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72319525 |
| maximal normal subgroup of | #72319525 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 291974);