Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511799 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×Q8 | C24 |
| prime radical | #3486 | C22 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
2514 conjugacy classes of subgroups; only the 192 normal subgroups are shown.
This group has 2514 conjugacy classes of subgroups — too many to draw. Only its 192 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:
(1,2)(3,4)(5,6)(7,15)(8,18)(9,13)(10,20)(11,16)(12,17)(14,19);(5,7,11,10)(6,12,16,18)(8,19,17,9)(13,15,14,20);(5,8,11,17)(6,13,16,14)(7,9,10,19)(12,20,18,15);(2,4)Polycyclic presentation:
2^9|b^2=i,c^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,h^a=hi,c^b=ci,e^b=ei,f^b=fhi,e^c=eh,f^c=fi,f^e=fi65 conjugacy classes — view the full character table.
| Fitting subgroup of | #72268125 |
| maximal subgroup of | #72268115, #72268125, #72268178, … |
| maximal normal subgroup of | #72268115, #72268125, #72268178, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 418996);