Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #17082 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2388 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3146 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6,8,10)(7,9,11,12)(13,14,16,18)(15,17,19,20)(21,22)(23,28)(24,25)(26,27);(5,7,8,11)(6,9,10,12)(13,15)(14,17)(16,19)(18,20)(21,23,26,25)(22,24,27,28);(6,10)(9,12)(15,19)(17,20)(21,24)(22,23)(25,27)(26,28);(2,4)(7,11)(9,12)(22,27)(24,28);(21,25,26,23)(22,24,27,28)Polycyclic presentation:
2^9|a^2=gh,b^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dfg,d^c=df92 conjugacy classes — view the full character table.
| maximal subgroup of | #72304079, #72304096, #72304129 |
| maximal normal subgroup of | #72304079, #72304096, #72304129 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 752129);