Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #458126 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #555 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2572 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
680 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)(7,17)(8,13,24,29)(9,16)(10,14)(11,15)(12,20)(18,32,34,36)(19,26)(21,27)(22,30,23,31)(25,33,28,35);(5,7)(6,12,15,27)(8,18)(9,20)(10,21)(11,19)(13,28,29,25)(14,17,16,26)(22,35)(23,33)(24,34)(30,32,31,36);(5,8,9,22,11,24,10,23)(6,13,14,31,15,29,16,30)(7,18,20,35,19,34,21,33)(12,25,17,32,27,28,26,36);(2,4)(6,14,15,16)(7,19)(8,22,24,23)(12,26,27,17)(13,29)(18,33,34,35)(20,21)(30,31)Polycyclic presentation:
2^9|a^2=fg,b^2=g,c^2=efg,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 | #72280214, #72280622 |
| maximal normal subgroup of | #72280214, #72280622 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 376457);