Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #125496 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1519 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×#97 | C23 |
| prime radical | C2×#469 | C4 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
369 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 369 conjugacy classes of subgroups — too many to draw. Only its 74 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:
(1,2,4,7,5,6,3,8)(9,10)(11,16,22,31)(12,24,27,15)(13,18)(14,20)(17,25,29,38)(19,26)(21,34)(23,32,36,33)(28,37)(30,40)(35,39);(2,6)(3,4)(10,15)(12,25)(13,26)(16,28)(17,19)(18,29)(20,24)(21,35)(23,36)(27,38)(30,33)(31,34)(32,39)(37,40);(1,3,5,4)(2,7,6,8)(9,11,14,22)(10,16,20,31)(12,21,27,37)(13,23,26,36)(15,28,24,34)(17,30,29,39)(18,32,19,33)(25,35,38,40)Polycyclic presentation:
2^9|a^2=e,c^2=hi,d^2=gi,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=cfg,d^a=df,g^a=gi,c^b=ch,d^b=dg,e^b=efgi,f^b=fi,g^b=gi,e^c=ei,e^d=ei50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 32762);