Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806903 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2279 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)(5,6)(7,10)(8,9)(11,12)(13,14,19,18)(15,20,24,27)(16,21,26,17)(22,25,28,23);(6,9)(7,11)(13,15,19,24)(14,20,18,27)(16,25,26,23)(17,22,21,28);(1,2,3,4)(13,16,18,17,19,26,14,21)(15,22,20,25,24,28,27,23);(6,9)(10,12)(16,26)(17,21)(22,28)(23,25);(5,7,8,11)(6,10,9,12)(13,17,18,26,19,21,14,16)(15,23,20,22,24,25,27,28)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=gh,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^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6723088);