Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24408 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3585 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3821 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)(7,15,17,11)(8,12)(9,14)(10,13)(16,25,34,33)(18,31,22,26)(19,27,35,36)(20,29)(21,28)(23,32)(24,30);(5,7)(6,11)(8,19)(9,22)(10,17)(12,27)(13,15)(14,31)(16,21)(18,24)(20,35)(23,34)(25,28)(26,30)(29,36)(32,33);(2,4)(5,8)(6,12)(7,16)(9,23)(10,20)(11,25)(13,29)(14,32)(15,33)(17,34)(18,35)(19,22)(21,24)(26,36)(27,31)(28,30);(5,7)(6,11)(8,16)(9,18)(10,17)(12,25)(13,15)(14,26)(19,21)(20,34)(22,24)(23,35)(27,28)(29,33)(30,31)(32,36)Polycyclic presentation:
2^9|a^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,f^b=fi,g^b=gi,h^b=hi,e^c=ei,g^c=gi,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72255896, #72280941, #72281136, … |
| maximal normal subgroup of | #72255896, #72280941, #72281136, … |
| maximal quotient of | #72313309, #72313483, #72313519, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266742);