Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12778 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1790 |
| commutator | C2×Q8 | C22×C8 |
| Frattini, cosocle | C2×#131 | C23 |
| socle | C2 | #8316 |
| prime radical | #3483 | C4 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C8 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
615 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 615 conjugacy classes of subgroups — too many to draw. Only its 110 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 — 3 generators on 32 points:
(1,2,6,13,7,15,22,31,8,14,21,30,5,12,20,28)(3,11,17,27,16,26,32,29,4,9,19,23,18,10,24,25);(2,9)(3,4)(6,21)(10,15)(11,14)(12,26)(13,23)(16,18)(20,22)(25,31)(27,30)(28,29);(1,3)(2,10)(4,8)(5,18)(6,19)(7,16)(9,12)(11,15)(13,29)(14,26)(17,21)(20,32)(22,24)(23,31)(25,30)(27,28)Polycyclic presentation:
2^9|a^2=f,d^2=i,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ci,d^b=di,e^b=eg,f^b=fgi,d^c=dgi,e^c=ei,f^c=fi,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72231593, #72231607, #72231620, … |
| maximal normal subgroup of | #72231593, #72231607, #72231620, … |
| maximal quotient of | #72239961, #72279609, C2×#129596, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36792);