Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153120 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13343 |
| commutator, Frattini, cosocle | C2×#127 | C23 |
| LCS | G ▷ C2×#127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×#127 ◁ G | |
| LECS | G ▷ C2×#127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
625 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 625 conjugacy classes of subgroups — too many to draw. Only its 50 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)(3,7)(4,8)(5,10)(6,11)(9,19)(12,28)(13,21)(14,23)(15,25)(16,26)(17,27)(18,22)(20,30)(24,29)(31,32);(1,3)(2,7)(4,12)(5,18)(6,13)(8,24)(9,22)(10,23)(11,21)(14,19)(15,29)(16,31)(17,30)(20,26)(25,28)(27,32);(1,4)(2,8)(3,12)(5,15)(6,17)(7,20)(9,26)(10,25)(11,27)(13,30)(14,29)(16,19)(18,31)(21,28)(22,24)(23,32)Polycyclic presentation:
2^9|g^2=i,b^a=bd,c^a=ce,f^a=fgh,h^a=hi,c^b=cf,e^b=ei,g^b=gh,d^c=dg,g^c=gi,h^c=hi,f^d=fhi,g^d=gi,g^f=gi32 conjugacy classes — view the full character table.
| maximal subgroup of | #72319450 |
| maximal normal subgroup of | #72319450 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60316);