Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #477321 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2830 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #17217 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
737 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)(3,11,13,7)(4,15)(5,20)(6,9)(8,19)(10,22)(12,28,17,32)(14,24,30,27)(16,21)(18,26)(23,29,25,31);(1,2)(3,7)(4,16,18,8)(5,20)(6,9)(10,22)(11,13)(12,23,17,25)(14,27)(15,21,26,19)(24,30)(28,31,32,29);(1,3)(2,7)(4,17,18,12)(5,14)(6,13)(8,25,16,23)(9,11)(10,24)(15,28,26,32)(19,31,21,29)(20,27)(22,30);(1,4)(2,8)(3,12)(5,21)(6,18)(7,23)(9,16)(10,26)(11,25)(13,17)(14,31)(15,20)(19,22)(24,28)(27,32)(29,30)Polycyclic presentation:
2^9|a^2=e,b^2=g,c^2=g,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=def,f^b=fi,h^b=hi,d^c=de,e^c=ei,g^d=gi,h^d=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386795);