Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4344 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,3,4)(9,10,12,17,13,16,22,15)(11,18,20,23,21,24,19,14);(5,6)(7,8)(9,11)(10,14)(12,19)(13,21)(15,18)(16,23)(17,24)(20,22);(2,4)(6,8);(2,4)(10,15)(11,19)(12,22)(16,17)(18,24)(20,21);(10,16)(14,23)(15,17)(18,24)Polycyclic presentation:
2^9|a^2=fg,f^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,f^b=fi,f^d=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72239646, #72284485, #72284692, … |
| maximal normal subgroup of | #72239646, #72284485, #72284692, … |
| maximal quotient of | #72239788, #72313075, #72313259, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7410531);