Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393979 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#392 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4069 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)(8,22)(9,14)(10,15)(11,16)(12,20)(13,23)(18,26)(19,27)(21,31)(24,30)(25,29)(28,34)(32,33)(35,36);(5,7)(6,12)(8,18)(9,19)(10,20)(11,21)(13,28)(14,17)(15,31)(16,27)(22,32)(23,33)(24,35)(25,34)(26,30)(29,36);(5,8)(6,13)(7,18)(9,23)(10,24)(11,25)(12,26)(14,29)(15,22)(16,30)(17,32)(19,33)(20,35)(21,34)(27,28)(31,36);(2,4)(5,9,11,10)(6,14,16,15)(7,19,21,20)(8,23,25,24)(12,17,27,31)(13,29,30,22)(18,33,34,35)(26,32,28,36)Polycyclic presentation:
2^9|d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,e^c=ei,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72260846, #72261238, #72264996, … |
| maximal normal subgroup of | #72260846, #72261238, #72264996, … |
| maximal quotient of | #72260879, #72262056, #72265592, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301177);