Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12715 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C2×#117 | C23 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#519 ◁ G | |
| LECS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
912 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 912 conjugacy classes of subgroups — too many to draw. Only its 122 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,4,7,5,8,3,6)(9,10)(11,19,22,34)(12,23,26,15)(13,17)(14,18)(16,21,32,28)(20,37)(24,36)(25,33,40,30)(27,35)(29,39)(31,38);(10,15)(13,27)(16,29)(17,33)(18,23)(19,36)(21,28)(25,40)(30,35)(31,32)(34,37)(38,39);(1,3,5,4)(2,6,8,7)(9,11)(10,16)(12,24)(13,28)(14,22)(15,29)(17,34)(18,32)(19,35)(20,26)(21,27)(23,31)(25,38)(30,36)(33,37)(39,40)Polycyclic presentation:
2^9|a^2=f,c^2=h,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,e^b=ei,f^b=fg,d^c=di,e^c=ei,f^c=fi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36528);