Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419843 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1003 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)(3,4)(5,6,11,15,9,12,10,23)(7,13,24,29,18,25,16,27)(8,19,22,26,20,30,21,14)(17,32,31,28,33,36,34,35);(6,12)(7,16,18,24)(8,20)(13,27,25,29)(15,23)(17,31,33,34)(21,22)(28,36,35,32);(5,7,9,18)(6,13)(8,17,20,33)(10,24,11,16)(12,25)(14,28)(15,27)(19,32)(21,31,22,34)(23,29)(26,35)(30,36);(2,4)(5,8,9,20)(6,14,12,26)(7,17,18,33)(10,22,11,21)(13,28,25,35)(15,30,23,19)(16,31,24,34)(27,36,29,32)Polycyclic presentation:
2^9|a^2=efg,b^2=ei,c^2=ef,d^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^c=ei,f^c=fi,g^c=gi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 342031);