Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #97690 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #379 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | C24 | C2×C22⋊C4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
820 conjugacy classes of subgroups; only the 154 normal subgroups are shown.
This group has 820 conjugacy classes of subgroups — too many to draw. Only its 154 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 52 points:
(5,6,9,14,10,13,17,20)(7,12,15,19,8,11,16,18)(21,22)(23,31)(24,35,37,28)(25,38)(26,30)(27,33)(29,41)(32,47,49,52)(34,45)(36,44,40,46)(39,43)(42,48,51,50);(1,2)(3,4)(5,7,10,8)(6,11,13,12)(9,16,17,15)(14,19,20,18)(21,23,26,34)(22,27,30,43)(24,32,37,49)(25,39,41,33)(28,42,35,51)(29,45,38,31)(36,50,40,48)(44,47,46,52);(2,4)(5,8,10,7)(6,12,13,11)(9,15,17,16)(14,18,20,19)(21,24,26,37)(22,28,30,35)(23,32,34,49)(25,40,41,36)(27,42,43,51)(29,46,38,44)(31,47,45,52)(33,50,39,48)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=h,d^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,d^b=di,e^b=ei,g^b=ghi,d^c=di,g^c=gh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 20253);