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) = C27×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 | |
3987 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)(5,6,8,10)(7,11,12,9)(13,14,16,19)(15,18,20,17)(21,22,25,28)(23,26,24,27);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,15)(14,17)(16,20)(18,19)(21,23,25,24)(22,26,28,27);(1,2,3,4)(6,10)(9,11)(15,20)(17,18);(5,7)(6,9)(8,12)(10,11)(13,14)(15,18)(16,19)(17,20)(21,22)(23,26)(24,27)(25,28);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,15)(14,18)(16,20)(17,19)(21,24,25,23)(22,27,28,26)Polycyclic presentation:
2^9|a^2=i,b^2=ghi,c^2=fgh,e^2=ghi,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, #72232373, #72232414, … |
| maximal normal subgroup of | #72230894, #72232373, #72232414, … |
| maximal quotient of | #72278160, #72278397, #72278569, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2427581);