Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #508292 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #32930 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
827 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14,18,25,24,28,23,19,17,22,15,20,16,21,26,27);(1,2)(3,4)(6,9)(10,12);(2,4)(5,7)(6,10)(8,11)(9,12)(14,19)(15,23)(16,24)(18,26)(20,21)(22,27)(25,28);(7,11)(10,12)Polycyclic presentation:
2^9|a^2=fi,f^2=hi,h^2=i,b^a=be,c^a=cf,c^b=cg,d^c=de,f^c=fh,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72343806, #72343851, #72343852, … |
| maximal normal subgroup of | #72343806, #72343851, #72343852, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 415492);