Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2805613 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 | |
3636 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)(5,6,8,11)(7,10,12,9)(13,14,16,18)(15,19,20,17)(22,25)(23,28);(5,7,8,12)(6,9,11,10)(15,20)(17,19)(21,22,24,25)(23,26,28,27);(13,15)(14,17)(16,20)(18,19)(21,23)(22,26)(24,28)(25,27);(1,2,3,4)(5,6)(7,9)(8,11)(10,12)(13,15)(14,17)(16,20)(18,19)(21,23)(22,27)(24,28)(25,26);(5,7)(6,10)(8,12)(9,11)(13,14)(15,17)(16,18)(19,20)(21,22)(23,26)(24,25)(27,28)Polycyclic presentation:
2^9|a^2=fg,b^2=f,d^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72232359, #72232369, #72232422, … |
| maximal normal subgroup of | #72232359, #72232369, #72232422, … |
| maximal quotient of | #72323542, #72330052, #72330333 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2831816);