Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2398 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(6,7)(8,12)(13,14)(15,20)(16,19)(17,18)(22,25)(23,27);(5,6,9,7)(8,11,12,10)(13,15,18,16)(14,19,17,20)(22,25)(26,28);(1,3,4,2)(5,6,9,7)(8,10,12,11)(13,16,18,15)(14,19,17,20)(21,22)(23,26)(24,25)(27,28);(5,7,9,6)(8,11,12,10)(21,23)(22,26)(24,27)(25,28);(5,8,9,12)(6,10,7,11)(13,17,18,14)(15,19,16,20)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fi,d^2=gh,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,e^b=ef128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 439812);