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 | |
2054 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(6,9)(7,11)(13,14,19,24)(15,25,16,21)(17,27,20,22)(18,23,26,28);(5,6)(7,10)(8,9)(11,12)(13,15,19,16)(14,21,24,25)(17,26,20,18)(22,28,27,23);(2,4)(13,16,20,18,19,15,17,26)(14,21,22,23,24,25,27,28);(5,7,8,11)(6,10,9,12)(13,17,19,20)(14,22,24,27)(15,26,16,18)(21,28,25,23);(6,9)(10,12)(13,18,17,16,19,26,20,15)(14,23,27,21,24,28,22,25)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=h,d^2=f,e^2=hi,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, 6715420);