Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #616960 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3782 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(6,9)(10,12)(13,14,16,17)(15,19,20,18)(21,22,25,26)(23,24,27,28);(14,17)(18,19)(22,26)(23,27);(1,2)(3,4)(21,23)(22,24)(25,27)(26,28);(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,17)(19,20)(21,23)(22,24)(25,27)(26,28);(5,7,8,11)(6,10,9,12)(13,15)(14,18)(16,20)(17,19)(21,24)(22,23)(25,28)(26,27)Polycyclic presentation:
2^9|a^2=fg,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=df92 conjugacy classes — view the full character table.
| maximal subgroup of | #72326993, #72327008, #72327053, … |
| maximal normal subgroup of | #72326993, #72327008, #72327053, … |
| maximal quotient of | #72335991, #72335993 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 541121);