Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23912 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#386 |
| prime radical | C2×#3449 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3896 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)(5,6,10,14,11,15,9,22)(7,12,17,28,18,27,23,26)(8,19,21,13,20,31,24,30)(16,32,35,29,33,36,34,25);(2,4)(5,7)(6,12)(8,16)(9,23)(10,17)(11,18)(13,29)(14,28)(15,27)(19,32)(20,33)(21,35)(22,26)(24,34)(25,30)(31,36);(5,8)(6,13)(7,16)(9,24)(10,21)(11,20)(12,25)(14,31)(15,30)(17,35)(18,33)(19,22)(23,34)(26,36)(27,29)(28,32);(2,4)(8,20)(13,30)(16,33)(19,31)(21,24)(25,29)(32,36)(34,35)Polycyclic presentation:
2^9|a^2=h,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,d^c=di,f^c=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72284680, #72284779, #72284902, … |
| maximal normal subgroup of | #72284680, #72284779, #72284902, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266667);