Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24986 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #555 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2495 |
| 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 | |
690 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,17)(8,22,26,30)(9,14)(10,15)(11,16)(12,20)(13,25,31,24)(18,34,35,27)(19,32)(21,28)(23,33,36,29);(1,2,3,4)(5,7,10,21,11,19,9,20)(6,12,14,32,16,28,15,17)(8,23,24,35,26,36,25,18)(13,29,22,27,31,33,30,34);(2,4)(5,8,9,24,11,26,10,25)(6,13,15,22,16,31,14,30)(7,18,20,23,19,35,21,36)(12,27,17,33,28,34,32,29);(6,14,16,15)(7,19)(12,17,28,32)(13,30,31,22)(18,35)(20,21)(23,36)(27,33,34,29)Polycyclic presentation:
2^9|a^2=h,b^2=efg,c^2=e,d^2=ef,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, 269889);