Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23995 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2598 |
| prime radical | C2×#2017 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1518 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6,10,15,11,16,9,14)(7,17,20,28,21,27,19,12)(8,22,25,30,23,31,24,13)(18,26,35,32,33,29,34,36);(1,2,4,3)(5,7,9,19,11,21,10,20)(6,12,15,17,16,28,14,27)(8,18,24,34,23,33,25,35)(13,29,22,36,30,26,31,32);(5,8)(6,13)(7,18)(9,24)(10,25)(11,23)(12,26)(14,31)(15,22)(16,30)(17,32)(19,34)(20,35)(21,33)(27,36)(28,29);(1,3,4,2)(6,14,16,15)(8,23)(12,27,28,17)(13,22,30,31)(18,33)(24,25)(26,32,29,36)(34,35)Polycyclic presentation:
2^9|a^2=h,b^2=e,d^2=eh,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,d^c=di,f^c=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72249991, #72257140, #72257201, … |
| maximal normal subgroup of | #72249991, #72257140, #72257201, … |
| maximal quotient of | #72265464 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 283340);