Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806903 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#127 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2182 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(6,9)(7,11)(13,14)(15,20)(16,24)(17,22)(18,23)(19,26)(21,27)(25,28);(5,6)(7,10)(8,9)(11,12)(13,15,18,26)(14,19,23,20)(16,25,27,17)(21,28,24,22);(1,2,3,4)(6,9)(10,12)(13,15,16,17,18,26,27,25)(14,20,24,22,23,19,21,28);(5,7,8,11)(6,10,9,12)(13,16,18,27)(14,21,23,24)(15,25,26,17)(19,28,20,22);(13,17,27,15,18,25,16,26)(14,22,21,20,23,28,24,19)Polycyclic presentation:
2^9|b^2=i,c^2=fghi,d^2=f,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6724464);