Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | D4×C4⋊C4 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1542 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)(5,6,10,17,12,18,11,16)(7,19,22,24,14,8,21,13)(9,25,27,31,28,32,26,15)(20,29,34,33,30,36,35,23);(5,7,10,22,12,14,11,21)(6,13,17,19,18,24,16,8)(9,20,27,34,28,30,26,35)(15,29,25,33,31,36,32,23);(2,4)(5,8,11,24,12,19,10,13)(6,14,17,21,18,7,16,22)(9,23,26,36,28,33,27,29)(15,30,25,35,31,20,32,34);(5,9)(6,15)(7,20)(8,23)(10,27)(11,26)(12,28)(13,29)(14,30)(16,32)(17,25)(18,31)(19,33)(21,35)(22,34)(24,36)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=g,e^2=i,f^2=i,g^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,e^b=ei,f^b=fi,g^b=gi,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 275345);