Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573615 |
| 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×#3524 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2994 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)(13,14,19,23,20,16,18,24)(15,17,22,21,26,28,25,27);(2,4)(5,6,8,9)(7,10,11,12)(13,15,20,26)(14,21,16,27)(17,24,28,23)(18,22,19,25);(6,9)(7,11)(13,16)(14,20)(15,21)(17,22)(18,24)(19,23)(25,28)(26,27);(5,7)(6,10)(8,11)(9,12)(13,17,20,28)(14,22,16,25)(15,23,26,24)(18,27,19,21);(6,9)(7,11)(13,18,20,19)(14,23,16,24)(15,25,26,22)(17,21,28,27)Polycyclic presentation:
2^9|a^2=fi,b^2=h,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=di,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6503723);