Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6351720 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
3601 conjugacy classes of subgroups; only the 566 normal subgroups are shown.
This group has 3601 conjugacy classes of subgroups — too many to draw. Only its 566 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 — 5 generators on 32 points:
(1,2)(3,7)(4,8)(5,10)(6,9)(11,13)(12,17)(14,21)(15,31)(16,23)(18,26)(19,24)(20,30)(22,32)(25,29)(27,28);(2,7)(3,13)(4,16)(8,20)(9,21)(10,11)(12,25)(14,28)(15,32)(17,19)(22,24)(23,31)(26,27)(29,30);(2,8)(3,14)(4,17)(7,20)(9,23)(10,24)(11,22)(12,26)(13,28)(15,29)(16,19)(21,31)(25,27)(30,32);(2,9)(7,21)(8,23)(10,26)(11,27)(12,24)(20,31)(22,25);(2,10)(7,11)(8,24)(9,26)(12,23)(20,22)(21,27)(25,31)Polycyclic presentation:
2^9|f^2=h,g^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,g^a=ghi,f^b=fh,g^b=gi,f^c=fi,g^c=ghi62 conjugacy classes — view the full character table.
| Fitting subgroup of | #72340787, #72340788 |
| maximal subgroup of | #72213691, #72219481, #72219529, … |
| maximal normal subgroup of | #72213691, #72219481, #72219529, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6258916);