Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61559 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C22 | C23×C4⋊C4 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1961 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(3,7)(6,8)(9,10,13,17,14,18,24,30)(11,15,21,27,20,26,33,37)(12,16,23,29,22,28,34,38)(19,25,32,36,31,35,39,40);(1,2,3,5,4,6,7,8)(9,11)(10,15)(12,19)(13,21)(14,20)(16,25)(17,27)(18,26)(22,31)(23,32)(24,33)(28,35)(29,36)(30,37)(34,39)(38,40);(9,12)(10,16)(11,19)(13,23)(14,22)(15,25)(17,29)(18,28)(20,31)(21,32)(24,34)(26,35)(27,36)(30,38)(33,39)(37,40);(12,22)(16,28)(19,31)(23,34)(25,35)(29,38)(32,39)(36,40);(11,20)(15,26)(19,31)(21,33)(25,35)(27,37)(32,39)(36,40)Polycyclic presentation:
2^9|a^2=g,b^2=f,f^2=h,g^2=i,b^a=bf,f^a=fh,e^b=ei,d^c=di122 conjugacy classes — view the full character table.
| maximal subgroup of | #72285711, C2×#6344427, #72286675, … |
| maximal normal subgroup of | #72285711, C2×#6344427, #72286675, … |
| maximal quotient of | C2×#6344427, #72287470, #72287534, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6251623);