Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #190218 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#381 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1194 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)(3,14,16,24)(4,17,19,8)(5,10)(6,22)(7,13)(9,18,27,31)(11,21)(12,20)(15,26,30,25)(23,28)(29,32);(1,3,5,4)(2,8,10,24)(6,15,20,18)(7,16,21,19)(9,12,25,22)(11,14,13,17)(23,30,32,31)(26,28,27,29);(1,4,5,3)(2,9,11,26)(6,15,20,18)(7,19,21,16)(8,12,14,28)(10,25,13,27)(17,29,24,22)(23,30,32,31);(2,10)(8,24)(9,25)(11,13)(12,22)(14,17)(26,27)(28,29)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=e,f^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,g^b=gh,f^c=fi,g^c=gh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72233291, #72233309, #72233312, … |
| maximal normal subgroup of | #72233291, #72233309, #72233312, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 97416);