Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #103161 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #392 |
| commutator, Frattini, cosocle | C24×C4 | C23 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C24×C4 ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C23 ▷ C1 | |
910 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 910 conjugacy classes of subgroups — too many to draw. Only its 88 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) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,24)(19,22,26,29)(20,23,27,31)(25,28,32,30);(2,6)(3,4)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19,21,26)(18,22,24,29)(20,25,27,32)(23,30,31,28);(1,3,5,4)(2,7,8,6)(13,16)(17,20)(18,23)(19,25)(21,27)(22,28)(24,31)(26,32)(29,30)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=g,d^2=g,b^a=bd,c^a=ce,d^a=dg,f^a=fhi,c^b=cf,d^b=dg,e^b=ei,d^c=dh56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10384);