Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23994 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2580 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
908 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(7,16,19,20)(12,17,26,30)(18,33,34,31)(25,35,36,32);(1,3,4,2)(5,6)(7,17,19,30)(8,13)(9,23)(10,14)(11,15)(12,20,26,16)(18,32,34,35)(21,28)(22,29)(24,27)(25,33,36,31);(5,7)(6,12)(8,18)(9,20)(10,16)(11,19)(13,25)(14,30)(15,26)(17,23)(21,31)(22,34)(24,33)(27,32)(28,35)(29,36);(5,8,10,21,11,22,9,24)(6,13,14,28,15,29,23,27)(7,18,16,31,19,34,20,33)(12,25,30,35,26,36,17,32)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^2=eg,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,f^b=fi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^d=fi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 278313);