Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32947 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2847 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14694 |
| prime radical | #60398 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3756 conjugacy classes of subgroups; only the 416 normal subgroups are shown.
This group has 3756 conjugacy classes of subgroups — too many to draw. Only its 416 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,7,12,5,8)(3,9,16,23,15,24,14,20)(4,10,19,25,18,26,17,21)(13,27,30,31,29,22,28,32);(2,8,12,11)(9,20,24,23)(10,21,26,25)(22,31,27,32);(1,3)(2,9)(4,13)(5,14)(6,16)(7,15)(8,20)(10,22)(11,23)(12,24)(17,28)(18,29)(19,30)(21,31)(25,32)(26,27);(1,4)(2,10)(3,13)(5,17)(6,19)(7,18)(8,21)(9,22)(11,25)(12,26)(14,28)(15,29)(16,30)(20,31)(23,32)(24,27)Polycyclic presentation:
2^9|a^2=h,b^2=eh,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,g^c=gi,f^d=fi92 conjugacy classes — view the full character table.
| Fitting subgroup of | #72341835 |
| maximal subgroup of | #72260677, #72260694, #72279536, … |
| maximal normal subgroup of | #72260677, #72260694, #72279536, … |
| maximal quotient of | #72260754, #72261031, C2×#392718, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299914);