Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2754 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,14)(15,17)(16,18)(19,20);(6,9)(10,12);(14,17)(15,20)(18,19);(1,3,4,2)(14,18)(17,19);(5,7,8,11)(6,10,9,12)Polycyclic presentation:
2^9|a^2=h,d^2=h,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,g^c=gi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72216614 |
| maximal normal subgroup of | #72216614 |
| maximal quotient of | #72216479 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6280478);