Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9091 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#386 |
| prime radical | C22×#575 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4398 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)(7,17,20,12)(8,13)(9,14)(10,16)(11,15)(18,33,35,26)(19,31,24,27)(21,36,34,28)(22,29)(23,30)(25,32);(5,7)(6,12)(8,21)(9,19)(10,24)(11,20)(13,28)(14,27)(15,17)(16,31)(18,22)(23,34)(25,35)(26,29)(30,36)(32,33);(2,4)(5,8)(6,13)(7,18)(9,22)(10,25)(11,23)(12,26)(14,29)(15,30)(16,32)(17,33)(19,34)(20,35)(21,24)(27,36)(28,31);(5,9,11,10)(6,14,15,16)(7,19,20,24)(8,22,23,25)(12,27,17,31)(13,29,30,32)(18,34,35,21)(26,36,33,28)Polycyclic presentation:
2^9|a^2=h,d^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,g^b=gi,h^b=hi,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72237503, #72237506, #72237510, … |
| maximal normal subgroup of | #72237503, #72237506, #72237510, … |
| maximal quotient of | #72267260, #72279662, #72281297, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264760);