Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32696 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2749 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
975 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,11,15,9,12,10,23)(7,13,19,28,18,24,16,26)(8,14,22,29,20,25,21,30)(17,27,31,35,32,34,33,36);(1,2)(3,4)(6,12)(7,16,18,19)(8,20)(13,26,24,28)(15,23)(17,31,32,33)(21,22)(27,35,34,36);(2,4)(5,7,9,18)(6,13)(8,17,20,32)(10,19,11,16)(12,24)(14,27)(15,26)(21,31,22,33)(23,28)(25,34)(29,36)(30,35);(5,8,9,20)(6,14,12,25)(7,17,18,32)(10,22,11,21)(13,27,24,34)(15,30,23,29)(16,31,19,33)(26,35,28,36)Polycyclic presentation:
2^9|a^2=h,b^2=ei,c^2=ef,d^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,g^d=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 291492);