Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454977 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23439 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1128 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)(7,16)(8,21)(9,13)(10,15)(11,18)(12,25)(14,26)(17,33)(19,28)(20,32)(22,27)(23,31)(24,30)(29,35)(34,36);(5,7)(6,11)(8,22,25,34)(9,19)(10,20)(12,29,21,33)(13,16)(14,32)(15,28)(17,24,35,23)(18,26)(27,31,36,30);(5,8,9,24,10,25,26,23)(6,12,14,30,15,21,13,31)(7,17,18,34,20,35,19,22)(11,27,16,29,28,36,32,33);(2,4)(6,13)(8,23)(9,26)(11,16)(12,21)(14,15)(17,22)(18,19)(24,25)(28,32)(29,33)(34,35)Polycyclic presentation:
2^9|b^2=f,c^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,e^b=ei,h^b=hi,d^c=de,f^c=fi,h^c=hi,e^d=ei,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 363030);