Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25798 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2433 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3021 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2283 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)(7,15,18,11)(8,20,21,12)(9,14)(10,13)(16,25,27,19)(17,26)(22,33,34,28)(23,24)(29,32)(30,35)(31,36);(1,2)(3,4)(5,7,9,19,10,18,24,25)(6,11,14,27,13,15,23,16)(8,17,22,31,21,32,34,35)(12,26,28,36,20,29,33,30);(2,4)(5,6)(7,16)(8,12)(9,23)(10,13)(11,25)(14,24)(15,19)(17,30)(18,27)(20,21)(22,33)(26,35)(28,34)(29,31)(32,36);(5,8)(6,12)(7,17)(9,22)(10,21)(11,26)(13,20)(14,28)(15,29)(16,30)(18,32)(19,31)(23,33)(24,34)(25,35)(27,36)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,f^b=fi,h^b=hi,e^c=ei,g^c=gi,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72281024, #72281044, #72281232, … |
| maximal normal subgroup of | #72281024, #72281044, #72281232, … |
| maximal quotient of | #72313270 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 273818);