Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #686200 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15123 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2758 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(5,6)(7,8)(9,10)(11,12)(13,14,17,19)(15,18,20,16)(23,27)(26,28);(1,2,3,4)(6,10)(8,11)(13,15)(14,16)(17,20)(18,19)(22,25)(26,28);(1,2)(3,4)(5,7,9,12)(6,8,10,11)(13,15,17,20)(14,16,19,18)(21,22)(23,26)(24,25)(27,28);(5,7)(6,11)(8,10)(9,12)(13,15)(14,18)(16,19)(17,20)(21,23,24,27)(22,26,25,28);(5,8,9,11)(6,7,10,12)(13,16,17,18)(14,15,19,20)Polycyclic presentation:
2^9|a^2=g,b^2=fhi,c^2=f,d^2=gh,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=df92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 605858);