Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153201 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13348 |
| commutator, Frattini, cosocle | C2×#124 | C23 |
| prime radical | C2×#2199 | C2 |
| LCS | G ▷ C2×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×#124 ◁ G | |
| LECS | G ▷ C2×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
558 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 558 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)(3,13,16,8)(4,7)(5,11)(6,10)(9,17)(12,21)(14,29,30,25)(15,26,20,28)(18,27,31,22)(19,23)(24,32);(2,7)(4,17)(8,25)(9,10)(11,23)(12,24)(13,29)(15,20)(18,31)(19,32)(22,26)(27,28);(1,3)(2,8)(4,18)(5,20)(6,16)(7,22)(9,27)(10,13)(11,26)(12,28)(14,19)(15,21)(17,31)(23,25)(24,29)(30,32)Polycyclic presentation:
2^9|a^2=h,d^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dhi,e^a=ei,f^a=fgh,g^a=gi,c^b=cf,d^b=dhi,g^b=gh,d^c=dg,g^c=gi,h^c=hi,f^d=fh,g^f=gi32 conjugacy classes — view the full character table.
| maximal subgroup of | #72219523, #72219528, #72219529, … |
| maximal normal subgroup of | #72219523, #72219528, #72219529, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60397);