Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×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 | |
4512 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)(7,10)(8,9)(11,12)(13,14)(15,19)(16,20)(17,18)(21,22)(23,28)(24,25)(26,27);(6,9)(10,12)(13,15)(14,16)(17,19)(18,20)(22,27)(23,24);(7,11)(10,12)(14,17)(15,20)(21,23,26,24)(22,28,27,25);(1,2)(3,4)(5,6)(7,12)(8,9)(10,11)(13,16)(14,15)(17,20)(18,19)(21,24,26,23)(22,25,27,28);(5,7,8,11)(6,10,9,12)(13,17,18,14)(15,19,20,16)(21,25)(22,24)(23,27)(26,28)Polycyclic presentation:
2^9|c^2=gh,d^2=gh,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
| maximal subgroup of | #72232597, #72232626, #72232627, … |
| maximal normal subgroup of | #72232597, #72232626, #72232627, … |
| maximal quotient of | #72237071, #72293980, #72293984, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3269497);