Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| 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 | |
3668 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)(6,7)(10,11)(13,14,16,17)(15,19,20,18)(22,25)(23,27);(5,6,9,7)(8,10,12,11)(13,14)(15,19)(16,17)(18,20)(23,27)(26,28);(8,12)(10,11)(14,17)(15,20)(21,22,24,25)(23,26,27,28);(1,2)(3,4)(5,7,9,6)(8,10,12,11)(13,14)(15,19)(16,17)(18,20)(21,22)(23,26)(24,25)(27,28);(5,8)(6,10)(7,11)(9,12)(13,15)(14,18)(16,20)(17,19)(21,23)(22,26)(24,27)(25,28)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=gh,d^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
| maximal subgroup of | #72234676, #72234679, #72305959, … |
| maximal normal subgroup of | #72234676, #72234679, #72305959, … |
| maximal quotient of | #72328991, #72329083, #72329091 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4052419);