Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23998 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2560 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)(7,11,17,26)(8,13,19,29)(9,12)(10,14)(15,22)(16,30)(18,23,21,25)(20,27,33,28)(24,32)(31,36)(34,35);(2,4)(5,7)(6,11,14,26)(8,16)(9,21)(10,17)(12,25,15,23)(13,24,29,30)(18,22)(19,32)(20,34)(27,36,28,35)(31,33);(6,12)(9,22)(11,23)(13,27)(14,15)(18,21)(20,33)(24,35)(25,26)(28,29)(30,36)(31,34);(2,4)(5,8,10,19)(6,13)(7,16,17,32)(9,20,22,33)(11,24)(12,27)(14,29)(15,28)(18,31,21,34)(23,35)(25,36)(26,30)Polycyclic presentation:
2^9|a^2=h,b^2=g,d^2=gi,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,f^c=fi,e^d=ei,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72306300, #72306319, #72306347, … |
| maximal normal subgroup of | #72306300, #72306319, #72306347, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271169);