Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60756 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#449 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2029 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6)(7,11)(8,10)(9,12)(13,14)(15,18)(16,23)(17,21)(19,22)(20,24)(25,27)(26,28);(5,7,8,12)(6,9,10,11)(13,15)(14,18)(16,23)(17,22)(19,21)(20,26)(24,28)(25,27);(7,12)(9,11)(13,16,19,25)(14,20,22,28)(15,23,21,27)(17,24,18,26);(6,10)(9,11)(13,14)(15,17)(16,20)(18,21)(19,22)(23,24)(25,28)(26,27);(1,3,4,2)(13,17,19,18)(14,21,22,15)(16,24,25,26)(20,27,28,23)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=cf,d^b=di,e^b=ei,g^c=gi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6285047);