Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417601 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60425 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1664 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6,9,14)(7,11,20,31)(8,21,22,12)(10,26,25,15)(13,23,33,24)(16,27,34,28)(17,35,36,32)(18,30,19,29);(1,2)(3,4)(5,7)(6,11)(8,17)(9,20)(10,19)(12,32)(13,28)(14,31)(15,30)(16,23)(18,25)(21,35)(22,36)(24,34)(26,29)(27,33);(5,8)(6,12)(7,16)(9,22)(10,24)(11,27)(13,15)(14,21)(17,19)(18,36)(20,34)(23,25)(26,33)(28,31)(29,35)(30,32);(1,3,4,2)(5,8)(6,13)(7,17)(9,22)(10,23)(11,28)(12,15)(14,33)(16,19)(18,34)(20,36)(21,26)(24,25)(27,31)(29,35)(30,32)Polycyclic presentation:
2^9|a^2=i,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,g^c=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325517);