Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #190055 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#381 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C25 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1046 conjugacy classes of subgroups; only the 250 normal subgroups are shown.
This group has 1046 conjugacy classes of subgroups — too many to draw. Only its 250 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 — 4 generators on 40 points:
(1,2,3,6,4,5,7,8)(9,10)(11,19,22,15)(12,23,24,38)(13,18)(14,17)(16,25,29,39)(20,33)(21,32,36,28)(26,30)(27,35)(31,40)(34,37);(9,11,14,22)(10,15,17,19)(12,20,24,34)(13,21,26,36)(16,27,29,31)(18,28,30,32)(23,37,38,33)(25,35,39,40);(2,5)(3,7)(9,12)(10,16,17,29)(11,20)(13,25)(14,24)(15,27,19,31)(18,23,30,38)(21,35)(22,34)(26,39)(28,37,32,33)(36,40);(12,24)(16,29)(20,34)(23,38)(25,39)(27,31)(33,37)(35,40)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^b=gh,d^c=dh,g^c=ghi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72317846, #72318396, #72318428 |
| maximal normal subgroup of | #72317846, #72318396, #72318428 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 97283);