Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438975 |
| 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 | C2×#3062 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2413 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)(5,6,8,10)(7,9,11,12)(14,20)(15,22)(17,18)(21,28)(23,24)(26,27);(1,2,3,4)(5,7)(6,9)(8,11)(10,12)(13,14,18,15,19,22,17,20)(16,24,27,28,25,21,26,23);(7,11)(9,12)(13,15,18,22,19,20,17,14)(16,23,27,24,25,28,26,21);(7,11)(9,12)(13,16)(14,21)(15,23)(17,26)(18,27)(19,25)(20,28)(22,24);(16,25)(21,24)(23,28)(26,27)Polycyclic presentation:
2^9|a^2=hi,b^2=fi,c^2=g,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6364554);