Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417682 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1489 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(6,12)(7,17)(8,22)(9,10)(13,18)(14,30)(15,16)(19,33)(20,21)(24,25)(28,36)(32,34);(1,3,4,2)(5,6)(7,18)(8,23)(9,15)(10,12)(11,16)(13,21)(14,22)(17,26)(19,34)(20,29)(24,30)(25,27)(28,35)(31,32)(33,36);(2,3)(5,7,11,21)(6,13,16,18)(8,19,25,33)(9,17,10,20)(12,26,15,29)(14,28,30,32)(22,35,24,31)(23,34,27,36);(5,8)(6,14)(7,19)(9,24)(10,22)(11,25)(12,27)(13,28)(15,23)(16,30)(17,31)(18,32)(20,35)(21,33)(26,36)(29,34)Polycyclic presentation:
2^9|a^2=ef,b^2=ef,c^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,e^b=ei,f^b=fi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,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, 328968);