Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
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 | |
3982 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)(7,11)(8,9)(10,12)(13,14,16,18)(15,17,19,20)(21,22)(23,24)(25,27)(26,28);(5,6)(7,11)(8,9)(10,12)(13,15,16,19)(14,17,18,20)(21,23,26,25)(22,27,28,24);(6,9)(10,11)(14,18)(17,20)(21,24,26,27)(22,23,28,25);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,17)(16,18)(19,20)(21,24)(22,23)(25,28)(26,27);(5,7)(6,10)(8,12)(9,11)(13,15)(14,17)(16,19)(18,20)(21,25,26,23)(22,27,28,24)Polycyclic presentation:
2^9|a^2=fgh,b^2=gh,c^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
| maximal subgroup of | #72323427, #72323435, #72323440, … |
| maximal normal subgroup of | #72323427, #72323435, #72323440, … |
| maximal quotient of | #72323559, #72333155 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5516120);