Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61071 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
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×#546 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2143 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(6,9)(10,12)(13,14,18,22)(15,17,21,19)(16,23,26,24)(20,27,28,25);(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,24)(17,22)(18,21)(20,25)(23,26)(27,28);(5,7,8,11)(6,10,9,12)(13,16,18,26)(14,20,22,28)(15,23,21,24)(17,25,19,27);(6,9)(10,12)(13,14)(15,17)(16,20)(18,22)(19,21)(23,25)(24,27)(26,28);(1,3,4,2)(13,17,18,19)(14,21,22,15)(16,25,26,27)(20,24,28,23)Polycyclic presentation:
2^9|a^2=h,c^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ci,d^b=df,e^b=ei,g^c=gi,e^d=ei110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6291337);