Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533195 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3036 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(13,14,20,26,19,24,18,25)(15,16,22,21,23,28,27,17);(5,6)(7,10)(8,9)(11,12)(13,15,19,23)(14,21,24,17)(16,25,28,26)(18,22,20,27);(2,4)(6,9)(10,12)(13,16,19,28)(14,22,24,27)(15,26,23,25)(17,20,21,18);(5,7)(6,10)(8,11)(9,12)(13,17,19,21)(14,23,24,15)(16,18,28,20)(22,26,27,25);(7,11)(10,12)(14,24)(16,28)(17,21)(25,26)Polycyclic presentation:
2^9|a^2=fgi,b^2=i,c^2=i,d^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,f^b=fi,d^c=di,e^c=ei,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6449587);