Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C25×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C2×#2403 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
5046 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6,9,10)(7,12,11,8)(13,14,16,18)(15,19,20,17)(21,22)(23,28)(24,25)(26,27);(6,10)(7,11)(13,15)(14,17)(16,20)(18,19)(21,23)(22,25)(24,26)(27,28);(1,3,4,2)(6,10)(7,11)(15,20)(17,19)(22,27)(23,24);(5,7,9,11)(6,8,10,12)(13,15)(14,17)(16,20)(18,19)(21,24,26,23)(22,25,27,28);(5,8)(6,11)(7,10)(9,12)(13,14)(15,17)(16,18)(19,20)(21,25)(22,23)(24,27)(26,28)Polycyclic presentation:
2^9|a^2=i,c^2=ghi,d^2=g,b^a=bf,c^a=cg,d^a=dgh,e^a=ef,c^b=ch,d^b=dg,d^c=df92 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#582893, #72293924, #72293925, … |
| maximal normal subgroup of | C2×#582893, #72293924, #72293925, … |
| maximal quotient of | #72293743, C2×#582893, #72293972, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 490089);