Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4311730 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×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 | |
3410 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)(5,6,9,8)(7,11,12,10)(13,14,16,19)(15,18,20,17)(21,22,26,28)(23,25,24,27);(6,8)(10,11)(13,15)(14,17)(16,20)(18,19)(21,23,26,24)(22,27,28,25);(7,12)(10,11)(15,20)(17,18)(22,28)(23,24);(1,2)(3,4)(5,7,9,12)(6,10,8,11)(13,15)(14,18)(16,20)(17,19)(21,24,26,23)(22,27,28,25);(5,8,9,6)(7,11,12,10)(13,14)(15,18)(16,19)(17,20)(21,25,26,27)(22,23,28,24)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=fgh,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,d^c=df80 conjugacy classes — view the full character table.
| maximal subgroup of | #72332348, #72332363, #72332369, … |
| maximal normal subgroup of | #72332348, #72332363, #72332369, … |
| maximal quotient of | #72332849 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4226039);