Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2236 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(5,6)(7,11)(8,12)(9,10)(14,18)(16,20);(6,10)(7,12)(14,18)(15,19)(21,22,24,26)(23,25,27,28);(6,10)(8,11)(13,14,17,18)(15,16,19,20)(21,22)(23,25)(24,26)(27,28);(5,7)(6,8)(9,12)(10,11)(13,15,17,19)(14,16,18,20)(21,23,24,27)(22,25,26,28);(5,8)(6,7)(9,11)(10,12)(13,16)(14,19)(15,18)(17,20)Polycyclic presentation:
2^9|a^2=i,b^2=fgh,c^2=fg,d^2=h,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dg,e^c=ef104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 484126);