Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500676 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#459 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1523 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(6,9)(7,12)(8,10)(13,14)(15,20)(16,19)(17,18);(1,2)(3,4)(5,6,10,13,11,14,8,9)(7,12,18,20,16,19,15,17);(2,4)(5,7)(6,12)(8,15)(9,17)(10,18)(11,16)(13,20)(14,19);(5,6,10,13,11,14,8,9)(7,12,18,20,16,19,15,17)Polycyclic presentation:
2^9|b^2=ei,d^2=ei,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,h^a=hi,c^b=cg,f^c=fh,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72214882, #72214886, #72214893, … |
| maximal normal subgroup of | #72214882, #72214886, #72214893, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407875);