Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23472 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#124 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#394 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1774 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)(8,19)(9,22)(10,14)(11,17)(12,20)(13,23)(16,28)(18,29)(21,27)(24,31)(25,32)(26,35)(30,36)(33,34);(5,7,9,17,10,18,23,32)(6,11,13,15,14,25,22,29)(8,16,21,31,20,30,34,35)(12,26,27,36,19,24,33,28);(5,8,9,21,10,20,23,34)(6,12,13,27,14,19,22,33)(7,16,17,31,18,30,32,35)(11,24,15,28,25,26,29,36);(8,20)(12,19)(16,30)(21,34)(24,26)(27,33)(28,36)(31,35)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=e,e^2=i,b^a=be,c^a=cf,e^a=ei,g^a=gi,c^b=cg,f^b=fi,d^c=di,f^c=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72317478, #72317483, #72317503 |
| maximal normal subgroup of | #72317478, #72317483, #72317503 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265260);