Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418854 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15266 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3027 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2001 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,21,7,8,6,11)(3,12,16,22,14,25,15,9)(4,17,18,10,19,26,20,23)(13,24,30,32,28,31,29,27);(2,8)(5,6)(9,22)(13,28)(15,16)(17,26)(18,20)(24,31);(1,3)(2,9)(4,13)(5,16)(6,15)(7,14)(8,22)(10,24)(11,25)(12,21)(17,27)(18,30)(19,28)(20,29)(23,31)(26,32);(1,4)(2,10)(3,13)(5,20)(6,18)(7,19)(8,23)(9,24)(11,26)(12,27)(14,28)(15,30)(16,29)(17,21)(22,31)(25,32)Polycyclic presentation:
2^9|a^2=ef,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,f^d=fi,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72285906, #72286231, #72286267, … |
| maximal normal subgroup of | #72285906, #72286231, #72286267, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326050);