Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #366639 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13995 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3019 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2330 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,7,10,5,20)(3,12,14,24,15,8,22,25)(4,16,19,9,18,27,21,26)(13,29,31,23,17,32,30,28);(1,3)(2,8)(4,17,18,13)(5,14)(6,22)(7,15)(9,23,26,28)(10,12)(11,24)(16,32,27,29)(19,31,21,30)(20,25);(1,4)(2,9)(3,13)(5,21)(6,19)(7,18)(8,23)(10,26)(11,27)(12,28)(14,31)(15,17)(16,20)(22,30)(24,29)(25,32);(2,10)(5,6)(8,12)(9,26)(14,22)(19,21)(23,28)(30,31)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,g^b=gi,h^b=hi,e^c=ei,f^c=fi,f^d=fi,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 273837);