Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33517 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C22 | C2×C4×D8 |
| prime radical | C2×#135 | C2×C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
471 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,3,5,6,8,7)(9,10,14,19)(11,21,24,38)(12,20,15,17)(13,26,28,34)(16,31,33,23)(18,25,36,29)(22,39,30,35)(27,37,40,32);(9,11)(10,16)(12,22)(13,27)(14,24)(15,30)(17,32)(18,35)(19,33)(20,37)(21,26)(23,29)(25,31)(28,40)(34,38)(36,39);(1,3,4,6,5,7,8,2);(9,12,14,15)(10,17,19,20)(11,22,24,30)(13,25,28,29)(16,32,33,37)(18,34,36,26)(21,35,38,39)(23,27,31,40)Polycyclic presentation:
2^9|a^2=f,c^2=fi,d^2=i,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=ch,d^a=di,e^a=eg,g^a=gi,e^b=eg,g^b=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72287259, #72287314, #72287321, … |
| maximal normal subgroup of | #72287259, #72287314, #72287321, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390983);