Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2676 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3465 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)(6,8)(14,18)(15,21)(16,23)(26,28);(1,2)(3,4)(9,10,11,12)(14,18)(15,21)(20,25)(22,27);(1,2,3,4)(9,10)(11,12)(13,14)(15,19)(16,20)(17,18)(21,24)(22,26)(23,25)(27,28);(5,6,7,8)(13,15,17,21)(14,19,18,24)(16,22,23,27)(20,26,25,28);(1,2)(3,4)(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)Polycyclic presentation:
2^9|b^2=gh,c^2=f,d^2=i,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72280883, #72280903, #72280957, … |
| maximal normal subgroup of | #72280883, #72280903, #72280957, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2213620);