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 | C2×#532 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2388 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)(7,11)(8,10)(9,12)(13,14,18,21,17,19,26,27)(15,22,24,25,23,20,28,16);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(14,19)(16,25)(20,22)(21,27);(7,12)(9,11)(13,15)(14,20)(16,21)(17,23)(18,24)(19,22)(25,27)(26,28);(6,10)(9,11)(14,19)(15,23)(21,27)(24,28);(13,16,18,22,17,25,26,20)(14,15,21,24,19,23,27,28)Polycyclic presentation:
2^9|a^2=g,b^2=h,e^2=g,g^2=i,b^a=bf,c^a=ci,d^a=df,c^b=cfi,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, 6277603);