Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C42 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator | C22 | C23×C42 |
| socle | C23 | C24×C4 |
| prime radical | C24 | C23×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2088 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,10)(8,11)(9,12)(13,14,19,22,18,23,17,27)(15,24,26,16,21,20,25,28);(1,2,3,4)(5,7,10,6)(8,11,9,12)(13,15)(14,20)(16,27)(17,25)(18,21)(19,26)(22,28)(23,24);(5,8,10,9)(6,11,7,12)(13,16,17,24,18,28,19,20)(14,21,27,26,23,15,22,25);(5,9,10,8)(6,11,7,12)(13,17,18,19)(14,22,23,27)(15,25,21,26)(16,20,28,24);(14,23)(15,21)(22,27)(25,26)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=fg,d^2=f,g^2=i,b^a=bf,d^a=df,e^a=ei,c^b=cf,e^b=ei,d^c=di152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6277527);