Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500186 |
| 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 | C2×#466 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1796 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(13,14,19,27,18,22,17,23)(15,21,26,20,16,25,24,28);(5,6)(7,10)(8,9)(11,12)(13,15,18,16)(14,20,22,28)(17,26,19,24)(21,23,25,27);(1,2,3,4)(6,9)(7,11)(15,16)(20,28)(21,25)(24,26);(5,7,8,11)(6,10,9,12)(13,16,18,15)(14,21,22,25)(17,26,19,24)(20,23,28,27);(6,9)(10,12)(14,22)(15,16)(23,27)(24,26)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=g,d^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,e^b=eh,f^b=fi,d^c=dh,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6427209);