Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12433 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #466 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C8×C4⋊C4 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
366 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 366 conjugacy classes of subgroups — too many to draw. Only its 114 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 44 points:
(2,4)(5,6,7,10,8,11,12,9)(13,14)(15,26)(16,31)(17,34)(18,36)(19,37)(20,25)(21,28)(22,32)(23,27)(24,35)(29,39)(30,44)(33,41)(38,42)(40,43);(1,2,3,4)(6,9)(7,12)(10,11)(13,15,27,42,19,30,17,28,20,16,32,43,35,33,18,29)(14,21,36,44,24,38,22,26,25,39,34,41,37,40,23,31);(13,16,32,42,19,33,18,28,20,15,27,43,35,30,17,29)(14,21,34,41,24,38,23,31,25,39,36,44,37,40,22,26)Polycyclic presentation:
2^9|a^2=f,b^2=dfghi,c^2=egi,d^2=ghi,e^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,c^b=ci,d^b=dh,f^b=fh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 48455);