Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24986 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #555 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2571 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
794 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,11,16)(7,17)(8,21,22,26)(9,12,10,15)(13,25)(14,24,30,23)(18,27)(19,28)(20,29)(31,32)(33,35)(34,36);(6,12,16,15)(7,18,20,25)(8,22)(13,27)(14,21,30,26)(17,29)(19,33,34,32)(23,24);(1,2,3,4)(5,7,11,20)(6,13)(8,19,22,34)(9,18,10,25)(12,17)(14,28)(15,29)(16,27)(21,35)(23,32,24,33)(26,31)(30,36);(5,8,9,23,11,22,10,24)(6,14,15,21,16,30,12,26)(7,19,18,32,20,34,25,33)(13,28,29,35,27,36,17,31)Polycyclic presentation:
2^9|a^2=h,b^2=eg,c^2=ef,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^b=di,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 269501);