Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511790 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #32898 |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2447 conjugacy classes of subgroups; only the 186 normal subgroups are shown.
This group has 2447 conjugacy classes of subgroups — too many to draw. Only its 186 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)(3,11)(4,8)(5,9)(6,10)(7,13)(12,24)(14,25)(15,21)(16,22)(17,30)(18,23)(19,26)(20,29)(27,32)(28,31);(1,3)(2,7)(4,12)(5,13)(6,14)(8,19)(9,11)(10,20)(15,26)(16,27)(17,28)(18,29)(21,24)(22,31)(23,25)(30,32);(1,4,6,16)(2,8,10,22)(3,12,14,27)(5,17,18,15)(7,19,20,31)(9,21,23,30)(11,24,25,32)(13,28,29,26);(5,18)(9,23)(11,25)(13,29)(15,17)(21,30)(24,32)(26,28)Polycyclic presentation:
2^9|c^2=i,b^a=be,c^a=cf,d^a=dg,h^a=hi,f^b=fhi,g^b=gi,e^c=eh,e^d=ei,f^e=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72218460, #72233445, #72233454, … |
| maximal normal subgroup of | #72218460, #72233445, #72233454, … |
| maximal quotient of | #72233416, #72235138, #72235143, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 418987);