Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C2×#2661 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2554 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)(5,6,9,8)(7,10,11,12)(13,14,17,18)(15,19,20,16)(22,25)(23,27);(6,8)(10,12)(13,15,17,20)(14,16,18,19)(21,22,24,25)(23,28,27,26);(7,11)(10,12)(14,18)(16,19)(21,23,24,27)(22,26,25,28);(5,7,9,11)(6,10,8,12)(13,16)(14,15)(17,19)(18,20)(21,23)(22,26)(24,27)(25,28);(5,8,9,6)(7,12,11,10)(13,15)(14,16)(17,20)(18,19)(21,22)(23,26)(24,25)(27,28)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=h,d^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=dh,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 492849);