Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #161894 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | #36624 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1740 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(6,9)(7,11)(10,12)(13,14,16,18,17,19,15,20);(5,6,7,9,8,10,11,12)(14,18)(15,16)(19,20);(2,4);(6,10)(9,12)(14,19)(18,20)Polycyclic presentation:
2^9|a^2=g,b^2=eg,e^2=hi,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,d^b=dh,e^b=ei,g^b=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72279310 |
| maximal normal subgroup of | #72279310 |
| maximal quotient of | #72279441 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 103888);