Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994107 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#127 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2319 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(7,11)(10,12)(13,14,16,20)(15,22,23,18)(17,19,25,21)(24,27,28,26);(2,3)(6,9)(7,11)(13,15)(14,18)(16,23)(17,24)(19,26)(20,22)(21,27)(25,28);(5,6)(7,10)(8,9)(11,12)(14,19)(17,25)(18,26)(20,21)(22,27)(24,28);(1,3,4,2)(6,9)(10,12)(15,23)(18,22)(24,28)(26,27);(5,7,8,11)(6,10,9,12)(14,20)(18,22)(19,21)(26,27)Polycyclic presentation:
2^9|a^2=f,d^2=fi,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7053037);