Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2031997 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×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 | |
3985 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)(8,12)(13,14)(15,19)(16,17)(18,20)(23,27)(25,28);(5,6,9,7)(8,11,12,10)(14,17)(15,20)(21,22,24,26)(23,25,27,28);(13,15)(14,18)(16,20)(17,19)(21,23)(22,25)(24,27)(26,28);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,15)(14,18)(16,20)(17,19)(21,22)(23,25)(24,26)(27,28);(5,8)(6,10)(7,11)(9,12)(13,14)(15,18)(16,17)(19,20)(21,23)(22,25)(24,27)(26,28)Polycyclic presentation:
2^9|b^2=fg,d^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72317852, #72317989, #72317997, … |
| maximal normal subgroup of | #72317852, #72317989, #72317997, … |
| maximal quotient of | #72319988, #72338552 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1942466);