Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2127 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C2×C4 | C22×C16 |
| Frattini, cosocle | C2×C4⋊C8 | C23 |
| socle | C22 | #1459 |
| prime radical | C23 | C2×C22⋊C8 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C8×Q8 ◁ G | |
| LECS | G ▷ C2×C4⋊C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
279 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 279 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) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,4,8,5,9,13,16,6,10,3,7,11,14,12,15)(17,18)(19,27,30,42)(20,24)(21,34,36,25)(22,26)(23,29)(28,43,31,45)(32,47,48,40)(33,41)(35,38)(37,44)(39,46);(17,19,22,30)(18,23,26,38)(20,31,33,28)(21,35,36,29)(24,39,41,37)(25,42,34,27)(32,44,48,46)(40,43,47,45);(1,3,5,12,6,4,11,13)(2,7,9,15,10,8,14,16)(17,20,22,33)(18,24,26,41)(19,28,30,31)(21,32,36,48)(23,37,38,39)(25,40,34,47)(27,43,42,45)(29,44,35,46)Polycyclic presentation:
2^9|a^2=e,b^2=h,c^2=gh,d^2=h,e^2=g,g^2=i,b^a=bd,d^a=df,f^a=fh,c^b=ch,d^b=dh,e^b=efh,e^d=eh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 29577);