Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394127 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1468 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6)(7,12)(8,21,23,30)(9,13)(10,15)(11,16)(14,22,28,24)(17,29)(18,32,34,35)(19,27)(20,25)(26,31,36,33);(1,2,4,3)(5,7)(6,12)(8,18)(9,19)(10,17)(11,20)(13,27)(14,26)(15,29)(16,25)(21,32)(22,31)(23,34)(24,33)(28,36)(30,35);(6,13)(7,17)(10,11)(12,25)(14,28)(18,31)(19,20)(22,24)(26,35)(27,29)(32,36)(33,34);(1,3,4,2)(5,8)(6,14)(7,18)(9,23)(10,22)(11,24)(12,26)(13,28)(15,21)(16,30)(17,31)(19,34)(20,33)(25,35)(27,36)(29,32)Polycyclic presentation:
2^9|a^2=fi,b^2=e,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gi,g^c=gi,h^c=hi,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 303465);