Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×#431 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
908 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,3)(5,6,8,11,9,7,10,12)(13,14,18,23)(15,19,25,28)(16,22,17,21)(20,26,27,24);(1,2,4,3)(5,7,8,12,9,6,10,11)(13,15)(14,19)(16,26)(17,24)(18,25)(20,21)(22,27)(23,28);(1,3,4,2)(13,15)(14,20)(16,24)(17,26)(18,25)(19,21)(22,28)(23,27);(13,16,18,17)(14,21,23,22)(15,24,25,26)(19,27,28,20)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=e,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fh,d^b=dh,f^c=fh128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 109526);