Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806551 |
| 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 | #2541 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#124 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2144 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)(5,6,9,11)(7,12,8,10)(13,14,18,19)(15,20,24,27)(16,25,21,17)(22,28,26,23);(5,6)(7,10)(8,12)(9,11)(13,15,18,24)(14,20,19,27)(16,23,21,28)(17,26,25,22);(1,2,3,4)(13,16,14,17,18,21,19,25)(15,22,27,28,24,26,20,23);(5,7,9,8)(6,10,11,12);(5,8,9,7)(6,10,11,12)(13,17,14,21,18,25,19,16)(15,23,27,22,24,28,20,26)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=gh,d^2=f,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6724042);