Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9103 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#386 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2578 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,17,20,12)(8,21,24,29)(9,14)(10,16)(11,15)(13,23,30,22)(18,32)(19,31,25,27)(26,33)(28,36)(34,35);(1,2)(3,4)(5,7)(6,12)(8,18)(9,19)(10,25)(11,20)(13,28)(14,27)(15,17)(16,31)(21,35)(22,33)(23,36)(24,34)(26,30)(29,32);(2,4)(5,8)(6,13,15,30)(7,18)(9,22)(10,23)(11,24)(12,26,17,28)(14,29,16,21)(19,33)(20,34)(25,36)(27,35,31,32);(5,9,11,10)(6,14,15,16)(7,19,20,25)(8,22,24,23)(12,27,17,31)(13,29,30,21)(18,33,34,36)(26,35,28,32)Polycyclic presentation:
2^9|a^2=h,c^2=e,d^2=i,f^2=i,b^a=be,c^a=cf,e^a=ei,g^a=gi,c^b=cg,f^b=fi,h^b=hi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72262939, #72262952, #72262980, … |
| maximal normal subgroup of | #72262939, #72262952, #72262980, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265072);