Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499266 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #3109 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #13316 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
593 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,8,15,24,13,31,30,10,14,23,29,9,16,22,12)(7,17,19,33,34,26,36,25,18,11,21,27,20,32,35,28);(1,2)(3,4)(5,7)(6,11)(8,21)(9,20)(10,18)(12,28)(13,32)(14,17)(15,33)(16,26)(19,23)(22,36)(24,34)(25,30)(27,29)(31,35);(2,4)(6,12)(8,22)(9,24)(11,25)(13,29)(14,30)(15,16)(17,28)(19,35)(20,34)(21,36)(23,31)(26,27)(32,33);(6,13,14,16)(7,18)(8,23)(11,26,17,32)(12,29,30,15)(20,34)(22,31)(25,27,28,33)Polycyclic presentation:
2^9|a^2=f,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=di,f^b=fi,f^c=fh,h^c=hi,f^d=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72239829, #72239836, #72239843, … |
| maximal normal subgroup of | #72239829, #72239836, #72239843, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406708);