Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417902 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | H(3,𝔽4) |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2804 |
| 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 | |
721 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)(5,6,9,16,11,15,10,14)(7,17,20,27,19,12,21,28)(8,13,24,30,23,31,25,32)(18,26,22,36,34,33,35,29);(5,7,9,21,11,19,10,20)(6,12,16,27,15,17,14,28)(8,22,24,18,23,35,25,34)(13,29,30,33,31,36,32,26);(5,8)(6,13)(7,18,19,34)(9,24)(10,25)(11,23)(12,26,17,33)(14,32)(15,31)(16,30)(20,22,21,35)(27,29,28,36);(2,4)(6,14,15,16)(8,23)(12,27,17,28)(13,30,31,32)(18,34)(22,35)(24,25)(26,36,33,29)Polycyclic presentation:
2^9|a^2=eg,b^2=efg,c^2=f,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,d^c=di62 conjugacy classes — view the full character table.
| maximal subgroup of | #72279655, #72280718, #72319908, … |
| maximal normal subgroup of | #72279655, #72280718, #72319908, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 348910);