Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C2×#2679 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4148 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,3)(9,10,12,13)(11,15,16,14)(18,19)(22,23);(6,8)(10,13)(11,16)(17,18,21,19)(20,23,24,22);(5,6,7,8)(10,13)(14,15)(18,19)(22,23);(1,3,4,2)(9,10)(11,14)(12,13)(15,16)(17,19,21,18)(20,23,24,22);(9,11,12,16)(10,14,13,15)(17,20)(18,22)(19,23)(21,24)Polycyclic presentation:
2^9|a^2=i,b^2=fg,c^2=fgh,d^2=fi,e^2=g,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df104 conjugacy classes — view the full character table.
| maximal subgroup of | #72329275, #72329361, #72329385, … |
| maximal normal subgroup of | #72329275, #72329361, #72329385, … |
| maximal quotient of | #72330085 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 470400);