Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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 | #2408 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2017 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,11)(8,10)(9,12)(13,14,18,22,17,19,26,27)(15,20,24,28,23,16,21,25);(5,7)(6,9)(8,12)(10,11)(14,19)(15,23)(21,24)(22,27);(1,2,3,4)(6,10)(9,11)(13,15,17,23)(14,20,19,16)(18,24,26,21)(22,28,27,25);(7,12)(9,11)(14,19)(15,23)(21,24)(22,27);(6,10)(9,11)(13,16,18,25,17,20,26,28)(14,21,22,15,19,24,27,23)Polycyclic presentation:
2^9|a^2=h,c^2=fg,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efi,c^b=ci,d^b=dfi,d^c=di,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6310733);