Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30496 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2669 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14405 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
923 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,11,7,12,6,10)(3,13,15,25,17,24,16,8)(4,18,20,28,22,27,21,9)(14,30,31,26,32,23,19,29);(1,3)(2,8)(4,19)(5,15)(6,16)(7,17)(9,26)(10,24)(11,13)(12,25)(14,20)(18,23)(21,32)(22,31)(27,30)(28,29);(1,4,6,21,7,22,5,20)(2,9,11,18,12,28,10,27)(3,14,16,19,17,32,15,31)(8,23,13,29,25,30,24,26);(3,15,17,16)(8,24,25,13)(14,31,32,19)(23,26,30,29)Polycyclic presentation:
2^9|a^2=h,c^2=f,d^2=ef,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 292623);