Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23439 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#390 |
| prime radical | C22×#575 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4370 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)(8,19,22,12)(9,14)(10,13)(16,26)(18,31,23,27)(20,28)(21,32,24,29)(25,30)(33,34)(35,36);(2,4)(5,7)(6,11)(8,20)(9,23)(10,17)(12,28)(13,15)(14,31)(16,21)(18,25)(19,36)(22,35)(24,34)(26,29)(27,30)(32,33);(5,8)(6,12)(7,16)(9,24)(10,22)(11,26)(13,19)(14,32)(15,33)(17,34)(18,35)(20,23)(21,25)(27,36)(28,31)(29,30);(7,17)(11,15)(16,34)(18,23)(20,35)(26,33)(27,31)(28,36)Polycyclic presentation:
2^9|a^2=h,g^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,c^b=cg,d^b=di,g^b=gi,h^b=hi,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72250486, #72250488, #72250505, … |
| maximal normal subgroup of | #72250486, #72250488, #72250505, … |
| maximal quotient of | #72258193, #72313731, #72313733, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264732);