Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806903 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2075 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)(15,19)(16,22)(17,21)(18,23)(20,25)(24,27)(26,28);(5,7)(6,9)(8,12)(10,11)(13,15,18,24)(14,19,23,27)(16,26,25,17)(20,21,22,28);(1,2,3,4)(13,16,17,15,18,25,26,24)(14,20,21,27,23,22,28,19);(6,10)(9,11)(13,17,18,26)(14,21,23,28)(15,16,24,25)(19,22,27,20);(7,12)(9,11)(13,16,17,15,18,25,26,24)(14,22,21,19,23,20,28,27)Polycyclic presentation:
2^9|b^2=i,c^2=ghi,d^2=i,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6721633);