Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533123 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3069 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)(14,19)(16,17)(20,26)(21,25)(23,24)(27,28);(5,6,10,9)(7,12,8,11)(13,14)(15,20)(16,19)(17,25)(18,21)(22,27)(23,26)(24,28);(2,4)(5,7,10,8)(6,11,9,12)(13,15)(14,20)(16,23)(17,24)(18,22)(19,26)(21,27)(25,28);(5,8,10,7)(6,11,9,12)(15,22)(20,27)(23,24)(26,28);(5,9,10,6)(7,12,8,11)(13,16,18,17)(14,19,21,25)(15,23,22,24)(20,26,27,28)Polycyclic presentation:
2^9|b^2=h,c^2=h,d^2=h,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,c^b=ch,f^b=fi,d^c=di,e^d=eh71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6479450);