Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11769 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1505 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×#97 | C23 |
| prime radical | C23 | #418 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #1726 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
282 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 282 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,4,7,5,8,3,6)(9,10)(11,21,23,37)(12,26,29,16)(13,19)(14,31)(15,20)(17,24,34,25)(18,30,36,27)(22,35)(28,33)(32,38)(39,40);(10,16,20,26)(12,27)(13,14)(17,32,34,40)(18,19,36,31)(21,33,37,35)(22,38)(24,25)(28,39)(29,30);(1,3,5,4)(2,6,8,7)(9,11)(10,17,20,34)(12,28)(13,24)(14,25)(15,23)(16,32,26,40)(18,35,36,33)(19,21,31,37)(22,29)(27,39)(30,38)Polycyclic presentation:
2^9|a^2=e,b^2=fg,c^2=fgh,d^2=g,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=cf,d^a=df,g^a=gi,d^b=dg,e^b=efg,f^b=fi,g^b=gi,d^c=di,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31698);