Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #96971 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #379 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | #35150 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1200 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 1200 conjugacy classes of subgroups — too many to draw. Only its 150 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 44 points:
(5,6)(7,11,12,9)(8,10)(13,14,18,23,19,24,17,22)(15,25,28,20,29,35,27,36)(16,30,31,21,33,37,32,38)(26,40,41,44,42,34,43,39);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,15)(14,20)(16,26)(17,28)(18,27)(19,29)(21,34)(22,35)(23,25)(24,36)(30,39)(31,43)(32,41)(33,42)(37,44)(38,40);(2,4)(5,7)(6,9)(8,12)(10,11)(13,16)(14,21)(15,26)(17,31)(18,32)(19,33)(20,34)(22,37)(23,30)(24,38)(25,39)(27,41)(28,43)(29,42)(35,44)(36,40)Polycyclic presentation:
2^9|a^2=g,d^2=i,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,d^b=di,g^b=ghi,e^c=ei,g^c=ghi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 18461);