Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| 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 | |
1838 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)(7,11)(8,10)(9,12)(13,14)(15,20)(16,18)(17,19);(5,6,8,10)(7,9,12,11)(14,16)(17,20)(22,25)(26,28);(1,3,4,2)(13,15)(14,17)(16,20)(18,19)(21,22)(23,26)(24,25)(27,28);(5,7)(6,9)(8,12)(10,11)(13,16,18,14)(15,20,19,17);(5,6)(7,9)(8,10)(11,12)(13,17,18,20)(14,19,16,15)(21,23,24,27)(22,26,25,28)Polycyclic presentation:
2^9|a^2=i,b^2=fg,c^2=i,d^2=g,e^2=gh,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=dg,e^b=ef104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 461351);