Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33605 |
| 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 ◁ C2×C4×C8 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
432 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 48 points:
(1,2,5,9,6,3,10,8)(4,7,12,15,13,11,16,14)(17,18)(19,28)(20,26)(21,33)(22,37)(23,25)(24,30)(27,35)(29,45)(31,44)(32,42)(34,40)(36,47)(38,43)(39,46)(41,48);(1,3)(2,6)(4,11)(5,8)(7,13)(9,10)(12,14)(15,16)(17,19)(18,24)(20,29)(21,34)(22,38)(23,31)(25,39)(26,40)(27,43)(28,33)(30,35)(32,46)(36,48)(37,41)(42,45)(44,47);(1,4,5,12,6,13,10,16)(2,7,9,15,3,11,8,14)(18,25)(24,39)(26,37)(27,42)(28,44)(33,47)(40,41)(43,45);(2,3)(7,11)(8,9)(14,15)(17,20,23,22)(18,26,25,37)(19,29,31,38)(21,32,36,35)(24,40,39,41)(27,33,42,47)(28,45,44,43)(30,34,46,48)Polycyclic presentation:
2^9|a^2=f,c^2=f,d^2=i,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=ci,d^a=dh,e^a=eg,g^a=gi,d^b=dh,e^b=eg,g^b=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72232014, #72232015, #72232016, … |
| maximal normal subgroup of | #72232014, #72232015, #72232016, … |
| maximal quotient of | #72286742, #72320205 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391140);