Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133835 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C23×Q8 | C23 |
| prime radical | C22×H(5,𝔽2) | C4 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#124 ◁ G | |
| LECS | G ▷ C23×Q8 ▷ C23 ▷ C2 ▷ C1 | |
1504 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 1504 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,7,8,5)(4,6)(9,10,13,17,14,18,29,36)(11,19,22,37,21,33,27,39)(12,23,26,35,25,16,28,34)(15,30,32,24,31,38,40,20);(9,11)(10,15)(12,24)(13,27)(14,21)(16,33)(17,32)(18,31)(19,23)(20,25)(22,29)(26,30)(28,38)(34,39)(35,37)(36,40);(1,3)(2,5)(4,8)(6,7)(12,25)(16,23)(19,33)(20,24)(26,28)(30,38)(34,35)(37,39)Polycyclic presentation:
2^9|a^2=f,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,d^b=di,e^b=ei,f^b=fgi,d^c=di,f^c=fh,f^d=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72227935, #72227937 |
| maximal normal subgroup of | #72227935, #72227937 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41033);