Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454389 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2815 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #16362 |
| prime radical | #60444 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1236 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,12,7,13,6,11)(3,14,16,8,18,26,17,10)(4,19,22,30,23,29,24,9)(15,25,20,27,32,31,21,28);(1,3,5,16,7,18,6,17)(2,8,11,14,13,10,12,26)(4,20,22,32,23,21,24,15)(9,27,29,25,30,28,19,31);(1,4)(2,9)(3,15)(5,24)(6,22)(7,23)(8,25)(10,31)(11,19)(12,29)(13,30)(14,27)(16,21)(17,20)(18,32)(26,28);(1,3,5,16,7,18,6,17)(2,10,11,26,13,8,12,14)(4,21,22,15,23,20,24,32)(9,28,29,31,30,27,19,25)Polycyclic presentation:
2^9|a^2=g,b^2=e,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,f^b=fi,d^c=de,e^c=ei,f^c=fi,g^c=gi,f^d=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72216037, #72216192, #72216253, … |
| maximal normal subgroup of | #72216037, #72216192, #72216253, … |
| maximal quotient of | #72237189, C2×#460333, #72255628, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 367529);