Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #456670 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2008 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1396 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,15,10,14,16,12)(7,11,19,25,17,27,20,30)(8,21,22,29,23,28,24,13)(18,32,33,26,31,35,34,36);(1,2)(3,4)(5,7,16,19,10,17,9,20)(6,11,15,30,14,27,12,25)(8,18,24,33,23,31,22,34)(13,26,21,32,29,36,28,35);(2,4)(6,12)(7,17)(9,16)(11,25)(13,21)(14,15)(22,24)(26,35)(27,30)(28,29)(32,36)(33,34);(5,8)(6,13)(7,18)(9,22)(10,23)(11,26)(12,28)(14,29)(15,21)(16,24)(17,31)(19,33)(20,34)(25,35)(27,36)(30,32)Polycyclic presentation:
2^9|a^2=f,b^2=efi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,d^c=de,f^c=fi,h^c=hi,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, 363868);