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 | #23422 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2431 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,9,11)(7,12,8,10)(13,14)(15,19)(16,22)(17,21)(18,23)(20,24)(25,28)(26,27);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15)(14,19)(16,26)(17,24)(18,25)(20,21)(22,27)(23,28);(2,4)(13,15)(14,20)(16,24)(17,26)(18,25)(19,21)(22,28)(23,27);(2,4)(5,8,9,7)(6,10,11,12);(13,16,18,17)(14,21,23,22)(15,24,25,26)(19,27,28,20)Polycyclic presentation:
2^9|a^2=f,b^2=f,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=dh,e^b=ei,g^c=gi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72284192, #72307379, #72307685, … |
| maximal normal subgroup of | #72284192, #72307379, #72307685, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346878);