Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #124690 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1548 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C43 | C23 |
| socle | C2 | #7609 |
| prime radical | C24 | #99 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C43 ▷ C22×C4 ▷ C2 ▷ C1 | |
402 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 402 conjugacy classes of subgroups — too many to draw. Only its 82 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)(3,12,15,7,14,21,18,22)(4,8)(5,9)(6,11)(10,19)(13,27,28,32,29,26,31,20)(16,24)(17,25)(23,30);(1,3,6,14)(2,7)(4,13,16,29)(5,18,19,15)(8,20)(9,12)(10,21)(11,22)(17,31,30,28)(23,27)(24,32)(25,26);(1,4)(2,8)(3,13)(5,17)(6,16)(7,20)(9,25)(10,23)(11,24)(12,26)(14,29)(15,28)(18,31)(19,30)(21,27)(22,32)Polycyclic presentation:
2^9|a^2=e,b^2=g,d^2=gi,e^2=h,f^2=i,b^a=bd,c^a=ch,d^a=df,f^a=fi,g^a=gi,d^b=dg,e^b=efg,f^b=fi,h^b=hi,d^c=di56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31886);