Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| 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 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2058 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)(3,4)(5,6,9,12)(7,8,11,10)(13,14,16,19)(15,21,23,18)(17,20,25,28)(22,27,24,26);(5,7,9,11)(6,10,12,8)(13,15,16,23)(14,18,19,21)(17,22,25,24)(20,26,28,27);(2,4)(5,8,9,10)(6,11,12,7)(15,22)(17,25)(18,26)(20,28)(21,27)(23,24);(2,4)(6,12)(7,11)(14,19)(15,23)(20,28)(22,24);(14,19)(18,21)(20,28)(26,27)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,h^b=hi,h^c=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72310827, #72310840, #72310857, … |
| maximal normal subgroup of | #72310827, #72310840, #72310857, … |
| maximal quotient of | #72313056, #72313239 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7409086);