Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator | C22 | C25×C4 |
| Frattini, socle, cosocle | C23 | C26 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7320 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2,3,4)(6,9)(7,11)(13,14)(15,18)(16,20)(17,19)(21,24)(22,25)(23,26)(27,28);(5,6)(7,10)(8,9)(11,12)(13,15)(14,18)(16,22)(17,21)(19,24)(20,25)(23,27)(26,28);(6,9)(10,12)(14,19)(18,24)(20,26)(25,28);(5,7,8,11)(6,10,9,12)(15,21)(18,24)(22,27)(25,28);(14,19)(18,24)(20,26)(25,28);(13,16,17,23)(14,20,19,26)(15,22,21,27)(18,25,24,28)Polycyclic presentation:
2^9|a^2=i,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=cg,d^b=dh152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7533197);