Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #478248 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2840 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1549 conjugacy classes of subgroups; only the 170 normal subgroups are shown.
This group has 1549 conjugacy classes of subgroups — too many to draw. Only its 170 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6)(7,10,16,19)(8,11)(9,13)(12,17)(14,20,15,18);(1,2)(3,4)(5,7)(6,10,13,19)(8,15)(9,16)(11,18,12,20)(14,17);(2,4)(5,8)(6,11)(7,14,16,15)(9,17)(10,18,19,20)(12,13);(5,8)(6,12,13,11)(7,15)(9,17)(10,18,19,20)(14,16)Polycyclic presentation:
2^9|a^2=f,b^2=ef,c^2=f,d^2=ef,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=def,e^b=ei,f^b=fi,h^b=hi,d^c=de,g^c=gi,h^c=hi,g^d=gi,h^d=hi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72237901, #72237922, #72237924, … |
| maximal normal subgroup of | #72237901, #72237922, #72237924, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 385445);