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×#2617 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3104 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(5,6)(7,8)(9,11)(10,12)(13,14)(15,19)(16,18)(17,20)(22,23)(24,28);(5,7,9,12)(6,10,11,8)(13,15,18,20)(14,17,16,19)(22,23)(26,27);(6,11)(8,10)(14,16)(15,20)(21,22,25,23)(24,26,28,27);(5,8,9,10)(6,7,11,12)(13,16,18,14)(15,17,20,19)(21,23,25,22)(24,27,28,26);(5,7)(6,10)(8,11)(9,12)(13,17)(14,15)(16,20)(18,19)(21,24)(22,26)(23,27)(25,28)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fg,d^2=g,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.
| maximal subgroup of | #72263742, #72263776 |
| maximal normal subgroup of | #72263742, #72263776 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 492478);