Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32696 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2749 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14379 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
953 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)(3,11,14,7)(4,15,18,29)(5,9)(6,10)(8,17,24,32)(12,27)(13,25,19,22)(16,30)(20,23)(21,28)(26,31);(1,3)(2,7)(4,16)(5,19)(6,14)(8,21)(9,25)(10,11)(12,18)(13,20)(15,27)(17,28)(22,23)(24,26)(29,30)(31,32);(1,4,5,17,6,18,20,32)(2,8,23,29,10,24,9,15)(3,12,13,28,14,16,19,31)(7,21,25,27,11,26,22,30);(2,9,10,23)(7,22,11,25)(8,15,24,29)(21,30,26,27)Polycyclic presentation:
2^9|a^2=h,c^2=ef,d^2=e,f^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,h^b=hi,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 291621);