Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
8809 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)(5,6)(7,9)(8,10)(11,12)(13,14)(15,18)(16,21)(17,20)(19,23)(22,25)(24,27)(26,28);(2,4)(5,7,8,11)(6,9,10,12)(13,15,18,14)(16,21,23,19)(17,22,25,20)(24,27,28,26);(6,10)(7,11)(13,16)(14,19)(15,21)(17,24)(18,23)(20,26)(22,27)(25,28);(6,10)(9,12)(14,15)(19,21)(20,22)(26,27);(16,23)(19,21)(24,28)(26,27);(13,17,18,25)(14,20,15,22)(16,24,23,28)(19,26,21,27)Polycyclic presentation:
2^9|b^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dh,c^b=ci,d^b=dhi,e^c=ehi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72279637, #72282475, #72284682, … |
| maximal normal subgroup of | #72279637, #72282475, #72284682, … |
| maximal quotient of | #72280171, #72281188, #72281212, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7548929);