Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| 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 | |
1986 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)(5,6)(7,8)(9,10)(11,12)(13,14)(15,20)(16,18)(17,19);(1,3,4,2)(6,10)(7,11)(14,16)(19,20)(22,25)(26,28);(13,15,18,17)(14,19,16,20)(21,22,24,25)(23,26,27,28);(5,7,9,11)(6,8,10,12)(13,16,18,14)(15,20,17,19)(21,23,24,27)(22,26,25,28);(5,8)(6,11)(7,10)(9,12)(13,17,18,15)(14,20,16,19)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=gh,d^2=fh,e^2=g,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 429538);