Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417622 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2408 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3010 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1875 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)(6,12)(7,17)(8,21)(9,11)(14,30)(15,16)(18,32)(19,20)(23,24)(25,28)(27,33)(35,36);(5,6,9,12,10,16,11,15)(7,13,17,28,20,29,19,25)(8,22,23,30,24,26,21,14)(18,33,34,27,32,35,31,36);(5,7)(6,13)(8,18)(9,19)(10,20)(11,17)(12,25)(14,27)(15,28)(16,29)(21,34)(22,33)(23,31)(24,32)(26,35)(30,36);(2,4)(5,8,10,24)(6,14,16,30)(7,18,20,32)(9,21,11,23)(12,26,15,22)(13,27,29,36)(17,31,19,34)(25,35,28,33)Polycyclic presentation:
2^9|b^2=f,d^2=i,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,g^b=gi,f^c=fi,h^c=hi,e^d=ei,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 328038);