Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2008 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #392 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1439 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
2478 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6)(7,17)(8,22,24,30)(9,15)(10,14)(11,16)(12,19)(13,23,31,25)(18,26,35,29)(20,27)(21,28)(32,34,36,33);(5,7,10,19,11,21,9,20)(6,12,14,28,16,27,15,17)(8,18,23,34,24,35,25,33)(13,29,22,32,31,26,30,36);(2,4)(5,8)(6,13)(7,18)(9,25)(10,23)(11,24)(12,26)(14,22)(15,30)(16,31)(17,32)(19,34)(20,33)(21,35)(27,29)(28,36)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=i,e^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,c^b=cf,d^b=di,d^c=di,e^c=ei,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72237503, #72237509, #72237511, … |
| maximal normal subgroup of | #72237503, #72237509, #72237511, … |
| maximal quotient of | #72239071, #72239086, #72239098, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30800);