Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #196278 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3449 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3950 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)(7,12)(8,20,22,31)(9,14)(10,15)(11,16)(13,21,29,24)(17,32,35,36)(18,26)(19,27)(23,30)(25,34,28,33);(2,4)(5,7,11,19)(6,12,16,27)(8,17,22,35)(9,18,10,23)(13,28,29,25)(14,26,15,30)(20,32,31,36)(21,33,24,34);(5,8)(6,13)(7,17)(9,21)(10,24)(11,22)(12,25)(14,20)(15,31)(16,29)(18,33)(19,35)(23,34)(26,36)(27,28)(30,32);(2,4)(5,9,11,10)(6,14,16,15)(7,18,19,23)(8,21,22,24)(12,26,27,30)(13,20,29,31)(17,33,35,34)(25,36,28,32)Polycyclic presentation:
2^9|a^2=h,b^2=i,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi,f^c=fi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72235458, #72251709, #72251718, … |
| maximal normal subgroup of | #72235458, #72251709, #72251718, … |
| maximal quotient of | #72235928, #72239270, #72239278, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266708);