Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #121999 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1446 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | C24 | #120 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
658 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 658 conjugacy classes of subgroups — too many to draw. Only its 142 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:
(2,5,6,7)(9,10)(11,20)(12,24,27,39)(13,28)(14,18)(15,19)(16,22)(17,26,34,25)(21,30,37,33)(23,31)(29,32)(35,36,38,40);(1,2)(3,7)(4,6)(5,8)(9,11)(10,16,19,32)(12,21)(13,22)(14,29)(15,23)(17,33,34,30)(18,20,28,31)(24,38,39,35)(25,36)(26,40)(27,37);(9,12)(10,17,19,34)(11,21)(13,25)(14,26)(15,27)(16,30,32,33)(18,24,28,39)(20,35,31,38)(22,36)(23,37)(29,40)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=f,d^2=hi,e^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,d^b=di,e^b=eh,g^b=gi,g^c=gh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72224700, #72224703, #72224704 |
| maximal normal subgroup of | #72224700, #72224703, #72224704 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 29195);