Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24805 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#386 |
| prime radical | #60444 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1797 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)(7,11)(8,12,19,29)(9,14)(10,13)(15,23,31,35)(16,24)(17,25)(18,26,33,36)(20,28,21,30)(22,27)(32,34);(2,4)(5,7)(6,11)(8,18,19,33)(9,17)(10,16)(12,26,29,36)(13,24)(14,25)(15,20,31,21)(22,32)(23,28,35,30)(27,34);(5,8,9,21,10,19,22,20)(6,12,27,28,13,29,14,30)(7,15,17,33,16,31,32,18)(11,23,34,26,24,35,25,36);(2,4)(7,16)(11,24)(15,31)(17,32)(18,33)(23,35)(25,34)(26,36)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=fg,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^b=di,f^c=fi,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72286985, #72286995, #72287097, … |
| maximal normal subgroup of | #72286985, #72286995, #72287097, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 275975);