Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418473 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2673 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15273 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1642 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)(3,12,15,25)(4,16)(5,10)(6,21)(7,11)(8,14,23,20)(9,18)(13,29,30,24)(17,32,31,26)(19,27)(22,28);(1,3)(2,8)(4,17)(5,20)(6,14)(7,15)(9,26)(10,25)(11,23)(12,21)(13,18)(16,29)(19,31)(22,30)(24,27)(28,32);(8,23)(12,25)(24,29)(26,32);(1,4,5,18,7,19,6,22)(2,9,10,27,11,28,21,16)(3,13,14,17,15,30,20,31)(8,24,12,26,23,29,25,32)Polycyclic presentation:
2^9|a^2=f,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72216013, #72216047, #72216142, … |
| maximal normal subgroup of | #72216013, #72216047, #72216142, … |
| maximal quotient of | #72237709, #72237713, C2×#419783, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326979);