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 | D4×C22⋊C4 |
| prime radical | #60643 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1812 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)(5,6,10,18,11,17,12,16)(7,19,21,13,22,30,8,14)(9,24,26,15,27,33,28,32)(20,31,25,34,35,29,23,36);(1,2,4,3)(5,7,11,22)(6,13)(8,10,21,12)(9,20,27,35)(14,17)(15,29)(16,19)(18,30)(23,26,25,28)(24,34)(31,32)(33,36);(5,8,10,7,11,21,12,22)(6,14,16,30,17,13,18,19)(9,25,26,35,27,23,28,20)(15,29,24,34,32,31,33,36);(1,3,4,2)(5,9,10,26,11,27,12,28)(6,15,16,24,17,32,18,33)(7,20,21,25,22,35,8,23)(13,29,19,34,14,31,30,36)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,g^a=gi,c^b=cg,f^b=fi,g^b=gi,d^c=di,f^c=fi,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 276402);