Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #458126 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #555 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2572 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
759 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,9,13,10,15,23,14)(7,16,19,11,18,29,34,25)(8,12,21,30,22,28,20,27)(17,31,32,26,33,36,35,24);(5,7)(6,11,15,25)(8,17)(9,19)(10,18)(12,26,28,24)(13,29,14,16)(20,35)(21,32)(22,33)(23,34)(27,31,30,36);(5,8)(6,12,15,28)(7,17)(9,20)(10,22)(11,24,25,26)(13,27,14,30)(16,31,29,36)(18,33)(19,35)(21,23)(32,34);(2,4)(6,13,15,14)(7,18)(8,20,22,21)(11,16,25,29)(17,32,33,35)(19,34)(24,26)(31,36)Polycyclic presentation:
2^9|a^2=eg,b^2=g,c^2=g,d^2=f,e^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,d^b=di,d^c=de,e^c=ei,f^c=fi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72280175, #72280622 |
| maximal normal subgroup of | #72280175, #72280622 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 376134);