Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #388397 |
| 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 | #23439 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1264 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,13)(3,14,19,30)(4,5)(6,26,8,12)(9,18,27,17)(10,22)(11,24)(15,28)(16,31)(20,32)(21,25)(23,29);(1,3)(2,9)(4,20)(5,23)(6,18)(7,19)(8,17)(10,28)(11,29)(12,30)(13,27)(14,26)(15,21)(16,24)(22,31)(25,32);(1,4,8,21,7,22,6,11)(2,10,26,25,13,5,12,24)(3,15,18,20,19,29,17,31)(9,16,30,23,27,32,14,28);(1,5,7,10)(2,11,13,21)(3,16,19,32)(4,12,22,26)(6,25,8,24)(9,20,27,31)(14,15,30,29)(17,28,18,23)Polycyclic presentation:
2^9|a^2=h,c^2=eh,d^2=i,e^2=i,g^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,f^b=fi,g^b=gi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,g^d=gi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295594);