Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #238077 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#393 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C26 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
1535 conjugacy classes of subgroups; only the 174 normal subgroups are shown.
This group has 1535 conjugacy classes of subgroups — too many to draw. Only its 174 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)(3,7)(4,15,18,8)(5,9)(6,11)(10,19)(12,27,16,24)(13,22)(14,23)(17,26,31,25)(20,28,32,30)(21,29);(1,3)(2,7)(4,16,18,12)(5,14)(6,13)(8,24,15,27)(9,23)(10,21)(11,22)(17,30,31,28)(19,29)(20,26,32,25);(1,4)(2,8)(3,12)(5,18)(6,17)(7,20)(9,15)(10,26)(11,25)(13,28)(14,16)(19,31)(21,27)(22,24)(23,32)(29,30);(4,17)(8,25)(12,28)(15,26)(16,30)(18,31)(20,24)(27,32)Polycyclic presentation:
2^9|a^2=e,b^2=e,b^a=be,c^a=cf,f^a=fh,g^a=ghi,c^b=cg,f^b=fi,g^b=gh,d^c=di,e^c=eh50 conjugacy classes — view the full character table.
| maximal subgroup of | #72211186, #72228236, #72228242, … |
| maximal normal subgroup of | #72211186, #72228236, #72228242, … |
| maximal quotient of | #72235133, #72235142, #72235231, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 145273);