Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6628872 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4865 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(7,13)(8,15)(10,17)(12,19)(14,16)(18,20);(1,2,3,4)(5,6)(7,10)(8,12)(9,11)(13,17)(14,18)(15,19)(16,20);(5,7)(6,10)(8,16)(9,14)(11,18)(12,20)(13,15)(17,19);(8,15)(12,19)(13,16)(17,20);(6,11)(10,18)(12,19)(17,20)Polycyclic presentation:
2^9|b^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,e^b=ei,g^c=gi,h^c=hi,g^d=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72219642, #72219652, #72219657, … |
| maximal normal subgroup of | #72219642, #72219652, #72219657, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536169);