Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #2424 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14,19,23,18,22,17,21)(15,24,25,27,26,20,28,16);(6,9)(10,12)(13,15)(14,20)(16,23)(17,28)(18,26)(19,25)(21,27)(22,24);(2,4);(13,16,19,24,18,27,17,20)(14,15,23,25,22,26,21,28);(5,7,8,11)(6,10,9,12)(13,17,18,19)(14,21,22,23)(15,25,26,28)(16,24,27,20)Polycyclic presentation:
2^9|a^2=h,d^2=h,e^2=f,h^2=i,b^a=bf,c^a=cg,e^b=ei,e^d=ei140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6280571);