Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437537 |
| 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 | C2×#3045 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2390 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)(5,6,8,9)(7,10,11,12)(13,14,19,23)(15,20,16,21)(17,26,18,22)(24,27,25,28);(6,9)(10,12)(15,24)(16,25)(17,18)(20,27)(21,28)(22,26);(2,4)(5,7,8,11)(6,10,9,12)(13,15)(14,20)(16,19)(17,24)(18,25)(21,23)(22,28)(26,27);(13,16)(14,21)(15,19)(17,25)(18,24)(20,23)(22,27)(26,28);(13,17,19,18)(14,22,23,26)(15,24,16,25)(20,28,21,27)Polycyclic presentation:
2^9|a^2=fi,c^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,e^b=ei,h^b=hi,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72284301, #72307406, #72307490, … |
| maximal normal subgroup of | #72284301, #72307406, #72307490, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345646);