Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| automorphism | Aut(G) order 1048576 |
| 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 | #15263 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2532 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)(6,7)(8,12)(14,17)(15,19)(22,26)(23,27);(5,6,9,7)(8,11,12,10)(14,17)(18,20)(21,22,25,26)(23,28,27,24);(6,7)(10,11)(13,14)(15,18)(16,17)(19,20)(21,23,25,27)(22,24,26,28);(1,2)(3,4)(5,7,9,6)(8,10,12,11)(13,14)(15,18)(16,17)(19,20)(21,23)(22,24)(25,27)(26,28);(5,8,9,12)(6,10,7,11)(13,15)(14,18)(16,19)(17,20)(21,24)(22,27)(23,26)(25,28)Polycyclic presentation:
2^9|b^2=f,c^2=fgh,d^2=gh,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2991727);