Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25443 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2361 |
| 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 | |
860 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)(5,6,11,16,12,18,10,17)(7,19,21,29,22,15,9,13)(8,23,26,14,27,31,25,32)(20,28,34,33,35,36,24,30);(2,4)(5,7,10,9,12,22,11,21)(6,13,16,19,18,29,17,15)(8,24,26,20,27,34,25,35)(14,30,23,36,32,33,31,28);(1,2,3,4)(5,8,11,26,12,27,10,25)(6,14,17,23,18,32,16,31)(7,20,21,34,22,35,9,24)(13,28,15,30,29,36,19,33);(5,9,10,22,12,21,11,7)(6,15,16,13,18,19,17,29)(8,24,25,35,27,34,26,20)(14,30,31,28,32,33,23,36)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,c^2=f,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,d^b=di,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 274768);