Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
5339 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)(7,9)(8,10)(11,12)(13,14,17,19)(15,20,16,18)(21,22,26,28)(23,24,25,27);(1,2)(3,4)(5,6,8,10)(7,9,11,12)(13,15,17,16)(14,18,19,20)(21,23,26,25)(22,27,28,24);(1,2,3,4)(7,11)(9,12)(15,16)(18,20)(22,28)(23,25);(5,6)(7,9)(8,10)(11,12)(13,14)(15,20)(16,18)(17,19)(21,24)(22,23)(25,28)(26,27);(1,2)(3,4)(5,7,8,11)(6,9,10,12)(13,16,17,15)(14,18,19,20)(21,25,26,23)(22,24,28,27)Polycyclic presentation:
2^9|a^2=i,b^2=fhi,c^2=hi,e^2=fhi,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 | #72230894, #72231231, #72232385, … |
| maximal normal subgroup of | #72230894, #72231231, #72232385, … |
| maximal quotient of | #72278109, #72278397, C2×#2563791, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2470987);