Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #496375 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #21810 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | #60647 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1095 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)(3,14,17,8)(4,19,20,10)(5,9)(6,24)(7,11)(12,16)(13,22)(15,30,18,27)(21,25,31,26)(23,28)(29,32);(1,3,6,18,7,17,22,15)(2,8,24,30,11,14,13,27)(4,16,21,23,20,32,31,5)(9,10,12,26,28,19,29,25);(2,9)(3,15)(5,16)(6,22)(8,25)(10,27)(11,28)(12,13)(14,26)(17,18)(19,30)(21,31)(23,32)(24,29);(1,4)(2,10)(3,16)(5,15)(6,21)(7,20)(8,12)(9,27)(11,19)(13,25)(14,29)(17,32)(18,23)(22,31)(24,26)(28,30)Polycyclic presentation:
2^9|a^2=g,b^2=eh,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,h^a=hi,c^b=ceh,g^b=gi,f^c=fh,h^c=hi,g^d=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403571);