Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #125300 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1505 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×#97 | C23 |
| prime radical | C22×D8 | C2×C4 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #1726 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
335 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 335 conjugacy classes of subgroups — too many to draw. Only its 80 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)(11,21,23,34)(12,25,29,15)(13,18)(14,20)(16,24,33,37)(17,26,31,40)(19,28)(22,35)(27,32)(30,39)(36,38);(2,6)(3,4)(10,15)(12,26)(13,28)(16,30)(17,19)(18,31)(20,25)(21,32)(22,36)(24,37)(27,39)(29,40)(33,38)(34,35);(1,3,5,4)(2,7,6,8)(9,11)(10,16,20,33)(12,27)(13,24)(14,23)(15,30,25,38)(17,35,31,32)(18,21,19,34)(22,29)(26,39)(28,37)(36,40)Polycyclic presentation:
2^9|a^2=e,c^2=fgi,d^2=gi,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=cf,d^a=df,g^a=gi,c^b=ch,d^b=dg,e^b=efgi,f^b=fi,g^b=gi,d^c=di,e^c=ei,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 32498);