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) = C24×C43 |
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 | |
4362 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)(7,8)(9,11)(10,12)(13,14,17,19)(15,16,20,18)(22,24)(23,27);(5,7,10,9)(6,11,12,8)(13,15)(14,18)(16,19)(17,20)(22,24)(26,28);(5,8,10,11)(6,7,12,9)(13,14,17,19)(15,18,20,16)(21,22,25,24)(23,28,27,26);(1,2)(3,4)(5,8)(6,7)(9,12)(10,11)(13,14)(15,18)(16,20)(17,19)(21,23,25,27)(22,26,24,28);(5,9,10,7)(6,11,12,8)(13,16,17,18)(14,15,19,20)(21,24,25,22)(23,28,27,26)Polycyclic presentation:
2^9|a^2=fgh,b^2=gh,c^2=fg,d^2=g,e^2=fg,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 | #72232325, #72232367, #72232383, … |
| maximal normal subgroup of | #72232325, #72232367, #72232383, … |
| maximal quotient of | #72278555, #72298139, #72298319, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5517016);