Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395191 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2431 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14825 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1218 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,12,8,14,6,13)(3,15,24,11,5,23,17,9)(4,18,22,28,21,27,20,10)(16,25,31,29,19,26,30,32);(1,3,6,17,8,5,7,24)(2,9,12,15,14,11,13,23)(4,19,20,31,21,16,22,30)(10,25,18,29,28,26,27,32);(1,4)(2,10)(3,16)(5,19)(6,20)(7,22)(8,21)(9,25)(11,26)(12,18)(13,27)(14,28)(15,29)(17,30)(23,32)(24,31);(1,5,6,24,8,3,7,17)(2,11,12,23,14,9,13,15)(4,19,20,31,21,16,22,30)(10,26,18,32,28,25,27,29)Polycyclic presentation:
2^9|a^2=eg,b^2=e,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,f^b=fi,f^c=fi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72216029, #72216192, #72216238, … |
| maximal normal subgroup of | #72216029, #72216192, #72216238, … |
| maximal quotient of | #72237187, C2×#401943, #72255722, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 309139);