Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #420381 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15277 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60454 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1351 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)(3,11,16,24)(4,10)(5,9,15,23)(6,12)(7,21)(8,13)(14,29,31,25)(17,32,30,22)(18,26)(19,27)(20,28);(1,3)(2,9)(4,17)(5,6)(7,15)(8,16)(10,25)(11,21)(12,24)(13,23)(14,19)(18,31)(20,30)(22,26)(27,32)(28,29);(1,4,6,18,8,20,7,19)(2,10,12,26,13,28,21,27)(3,14,15,30,16,31,5,17)(9,22,11,25,23,32,24,29);(1,5,7,16,8,15,6,3)(2,11,21,9,13,24,12,23)(4,17,19,31,20,30,18,14)(10,25,27,22,28,29,26,32)Polycyclic presentation:
2^9|a^2=f,c^2=e,d^2=ei,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^c=hi,f^d=fi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72215936, #72216013, #72216029, … |
| maximal normal subgroup of | #72215936, #72216013, #72216029, … |
| maximal quotient of | C2×#420382, #72255748, #72255797, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327578);