Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4307937 |
| automorphism | Aut(G) order 16777216 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C24×C43 |
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 | |
2432 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)(13,14,17,20)(15,16,19,18)(23,24)(27,28);(13,15,17,19)(14,18,20,16)(22,26)(27,28);(1,2,3,4)(13,14)(15,18)(16,19)(17,20)(21,22)(23,27)(24,28)(25,26);(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,19)(17,20)(21,23,25,24)(22,27,26,28);(5,7,8,11)(6,10,9,12)(13,16,17,18)(14,19,20,15)(21,24,25,23)(22,28,26,27)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=fg,d^2=h,e^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4215894);