Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24292 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2366 |
| prime radical | C2×#3150 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1818 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,11,16,12,18,10,17)(7,19,22,30,23,33,21,13)(8,14,25,32,26,15,9,27)(20,31,24,36,35,29,34,28);(5,7)(6,13,18,30)(8,24)(9,20)(10,21)(11,22)(12,23)(14,31,15,29)(16,19,17,33)(25,35)(26,34)(27,28,32,36);(1,2,3,4)(5,8,10,9,12,26,11,25)(6,14,17,27,18,15,16,32)(7,20,22,24,23,35,21,34)(13,28,19,31,30,36,33,29);(5,9,10,26,12,25,11,8)(6,15,16,27,18,14,17,32)(7,20,21,34,23,35,22,24)(13,29,19,28,30,31,33,36)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=fg,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 276366);