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) = C27 |
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 | |
2760 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,12)(8,9)(10,11)(13,14,17,20)(15,16,19,18)(22,26)(23,28);(2,4)(5,7)(6,9)(8,10)(11,12)(13,15,17,19)(14,18,20,16)(21,22)(23,24)(25,26)(27,28);(1,2)(3,4)(6,11)(7,8)(15,19)(16,18)(21,23,25,28)(22,27,26,24);(5,8,10,7)(6,9,11,12)(13,16,17,18)(14,19,20,15)(21,23)(22,27)(24,26)(25,28);(5,9)(6,8)(7,11)(10,12)(13,14)(15,16)(17,20)(18,19)(21,24,25,27)(22,28,26,23)Polycyclic presentation:
2^9|a^2=fg,b^2=fg,c^2=i,d^2=fi,e^2=i,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, 3207252);