Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7389 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22 | #1333 |
| commutator | C16 | C2×C16 |
| Frattini, cosocle | #1311 | C22 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C82 ◁ #1311 ◁ G | |
| LECS | G ▷ #1311 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
88 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 2 generators on 48 points:
(1,2,3,6,4,7,9,12,5,8,10,13,11,14,15,16)(18,23,24,38,26,40,27,41)(19,28,31,43)(20,33)(21,35)(25,29,34,45,37,46,42,39)(30,47)(32,36,48,44);(17,18,22,26)(19,29,31,46)(20,34,30,42)(21,24,35,27)(23,36,40,44)(25,33,37,47)(28,38,43,41)(32,39,48,45)Polycyclic presentation:
2^9|a^2=d,b^2=i,c^2=ei,d^2=f,e^2=g,f^2=h,g^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gi,c^b=ceg,d^b=dg,e^b=egi,f^b=fi,g^b=gi,d^c=di68 conjugacy classes — view the full character table.
| maximal subgroup of | #72286442, #72286459, #72286494, … |
| maximal normal subgroup of | #72286442, #72286459, #72286494, … |
| maximal quotient of | #72286530 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 954);