Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #772778 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2560 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(6,9)(10,12)(13,14,18,17)(15,19,16,20)(21,22,25,27)(23,24,26,28);(13,14)(15,20)(16,19)(17,18)(21,22)(23,24)(25,27)(26,28);(1,2)(3,4)(13,15,18,16)(14,19,17,20)(21,23,25,26)(22,24,27,28);(5,6)(7,10)(8,9)(11,12)(13,16,18,15)(14,19,17,20)(21,22)(23,24)(25,27)(26,28);(5,7,8,11)(6,10,9,12)(13,17,18,14)(15,20,16,19)(21,24)(22,26)(23,27)(25,28)Polycyclic presentation:
2^9|a^2=f,c^2=f,d^2=h,e^2=fhi,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,e^d=ef98 conjugacy classes — view the full character table.
| maximal subgroup of | #72280373 |
| maximal normal subgroup of | #72280373 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 746611);