Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #388379 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14519 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23400 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1295 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,8,13,6,11,7,12)(3,9,18,29,16,30,17,28)(4,5,21,24,20,23,10,25)(14,31,19,15,27,26,32,22);(1,3)(2,9)(4,19)(5,22)(6,16)(7,18)(8,17)(10,27)(11,30)(12,29)(13,28)(14,21)(15,23)(20,32)(24,26)(25,31);(1,4,7,21,6,20,8,10)(2,5,13,25,11,23,12,24)(3,14,17,19,16,27,18,32)(9,26,29,15,30,31,28,22);(1,5)(2,10)(3,15)(4,13)(6,23)(7,25)(8,24)(9,27)(11,21)(12,20)(14,30)(16,22)(17,31)(18,26)(19,28)(29,32)Polycyclic presentation:
2^9|a^2=h,c^2=eh,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,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,g^d=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295577);