Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #137661 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1893 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#469 | C4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
413 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 413 conjugacy classes of subgroups — too many to draw. Only its 98 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:
(1,2,5,7,4,3,6,8)(9,10,14,19,15,18,33,20)(11,21,17,32,23,36,24,13)(12,25,28,38,30,16,29,35)(22,39,34,31,26,40,37,27);(1,3,5,8,4,2,6,7)(9,11,13,10,14,24,21,20,15,23,32,18,33,17,36,19)(12,26,31,25,29,37,39,38,30,22,27,16,28,34,40,35);(2,3)(7,8)(9,12)(10,16)(11,22)(13,31)(14,29)(15,30)(17,37)(18,25)(19,38)(20,35)(21,39)(23,26)(24,34)(27,32)(28,33)(36,40)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,d^2=g,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=ei,f^b=fi,d^c=di,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72237243, #72239582, #72239584 |
| maximal normal subgroup of | #72237243, #72239582, #72239584 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44858);