Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573591 |
| 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 |
| prime radical | C2×#3515 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2976 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)(13,14,18,22,20,24,19,23)(15,25,27,17,16,28,26,21);(2,4)(5,6,8,9)(7,10,11,12)(13,15)(14,21)(16,20)(17,24)(18,26)(19,27)(22,28)(23,25);(6,9)(7,11)(13,16,20,15)(14,21,24,17)(18,27,19,26)(22,28,23,25);(5,7)(6,10)(8,11)(9,12)(13,17,20,21)(14,16,24,15)(18,28,19,25)(22,26,23,27);(6,9)(7,11)(13,18,20,19)(14,22,24,23)(15,26,16,27)(17,25,21,28)Polycyclic presentation:
2^9|a^2=f,b^2=hi,c^2=i,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=ehi,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=eh83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6503483);