Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #481415 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#102 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| prime radical | C2×#41 | SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ C8×#41 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
707 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 48 points:
(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(17,18,21,25,22,27,37,26)(19,28,30,41,31,45,36,44)(20,32,34,24,35,42,48,43)(23,33,38,29,40,46,39,47);(17,19,22,31)(18,23,27,40)(20,33,35,46)(21,36,37,30)(24,41,43,44)(25,39,26,38)(28,32,45,42)(29,34,47,48);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16);(3,9)(6,12)(10,15)(13,16)Polycyclic presentation:
2^9|a^2=f,b^2=i,e^2=g,f^2=h,g^2=i,b^a=be,e^a=eg,g^a=gi,e^b=egi,f^b=fi,g^b=gi,d^c=dhi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72287554, #72287638, #72287643, … |
| maximal normal subgroup of | #72287554, #72287638, #72287643, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 388620);