Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511793 |
| 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 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2454 conjugacy classes of subgroups; only the 186 normal subgroups are shown.
This group has 2454 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)(12,27)(14,26)(15,21)(16,22)(17,24)(18,23)(19,29)(20,30)(25,28)(31,32);(1,3)(2,7)(4,12)(5,13)(6,14)(8,19)(9,11)(10,20)(15,28)(16,29)(17,30)(18,31)(21,32)(22,27)(23,25)(24,26);(4,15)(5,17)(8,21)(12,28)(13,30)(19,32)(22,23)(25,27);(1,4,6,15)(2,8,10,21)(3,12,14,28)(5,18,17,16)(7,19,20,32)(9,23,24,22)(11,25,26,27)(13,31,30,29)Polycyclic presentation:
2^9|d^2=i,b^a=be,c^a=cf,d^a=dg,h^a=hi,f^b=fhi,g^b=gi,d^c=di,e^c=eh,e^d=ei,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72233205, #72240648, #72249655, … |
| maximal normal subgroup of | #72233205, #72240648, #72249655, … |
| maximal quotient of | #72219533, #72233099, #72233152, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 418990);