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 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3407 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)(3,4)(9,10,14,17,15,13,12,21)(11,18,19,22,20,24,23,16);(5,6)(7,8)(9,11)(10,16)(12,19)(13,22)(14,23)(15,20)(17,24)(18,21);(2,4)(6,8)(9,12,15,14)(10,17,13,21)(11,19,20,23)(16,24,22,18);(2,4)(9,13,15,10)(11,18,20,24)(12,21,14,17)(16,23,22,19);(11,20)(16,22)(18,24)(19,23)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,e^b=ei,f^b=fi,f^d=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72315311, #72315321, #72315341, … |
| maximal normal subgroup of | #72315311, #72315321, #72315341, … |
| maximal quotient of | #72315860 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7408184);