Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33830 |
| 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 | #2215 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ #431 ◁ C8×M16 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
414 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 48 points:
(1,2,6,9,5,8,14,16)(3,10,12,13,11,7,15,4)(17,18,23,27,24,29,21,28)(19,30,32,42,35,48,34,47)(20,36,22,39,38,26,37,46)(25,41,44,31,45,40,43,33);(1,3)(2,7)(4,9)(5,11)(6,12)(8,10)(13,16)(14,15)(17,19,24,35)(18,25,29,45)(20,31,38,33)(21,34,23,32)(22,40,37,41)(26,42,36,47)(27,44,28,43)(30,46,48,39);(1,4,6,10,5,13,14,7)(2,3,9,12,8,11,16,15)(17,20,23,22,24,38,21,37)(18,26,27,46,29,36,28,39)(19,31,32,40,35,33,34,41)(25,42,44,48,45,47,43,30);(3,11)(4,13)(7,10)(12,15)(17,21,24,23)(18,27,29,28)(19,32,35,34)(20,37,38,22)(25,43,45,44)(26,46,36,39)(30,47,48,42)(31,40,33,41)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=f,d^2=i,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=ci,d^a=di,e^a=eg,g^a=gi,d^b=dh,e^b=eg,g^b=gi,d^c=dhi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72234320, #72234324, #72234328, … |
| maximal normal subgroup of | #72234320, #72234324, #72234328, … |
| maximal quotient of | #72320179, #72320193 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391326);