Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470355 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2015 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)(7,11)(10,12)(13,14,19,24)(15,25,26,28)(16,21,20,27)(17,23,18,22);(5,6)(7,10)(8,9)(11,12)(13,15,17,27,19,26,18,21)(14,20,22,28,24,16,23,25);(2,4)(6,9)(10,12)(13,16)(14,21)(15,22)(17,25)(18,28)(19,20)(23,26)(24,27);(6,9)(10,12)(15,26)(16,20)(21,27)(25,28);(5,7,8,11)(6,10,9,12)(13,17,19,18)(14,22,24,23)(15,27,26,21)(16,25,20,28)Polycyclic presentation:
2^9|a^2=i,b^2=fi,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6398897);