Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574270 |
| 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 |
| prime radical | #38034 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2109 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(13,14,19,22)(15,20,18,16)(17,21,25,28)(23,27,26,24);(2,4)(5,6,8,9)(7,10,11,12)(13,15,19,18)(14,20,22,16)(17,23,25,26)(21,27,28,24);(6,9)(7,11)(13,16,18,14,19,20,15,22)(17,24,26,21,25,27,23,28);(5,7)(6,10)(8,11)(9,12)(13,17)(14,21)(15,23)(16,24)(18,26)(19,25)(20,27)(22,28);(6,9)(7,11)(17,25)(21,28)(23,26)(24,27)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=g,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ehi,d^c=dhi,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6513283);