Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32696 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2749 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
641 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6,9,15)(7,11)(8,20)(10,14,16,13)(12,27)(17,32,33,36)(18,26)(19,25)(21,29)(22,30)(23,31)(24,34,28,35);(5,7,16,19,9,18,10,23)(6,11,13,25,15,26,14,31)(8,17,22,35,21,33,27,34)(12,28,20,32,30,24,29,36);(1,3,4,2)(5,8,16,27,9,21,10,22)(6,12,14,20,15,30,13,29)(7,17,23,35,18,33,19,34)(11,24,25,32,26,28,31,36);(6,13,15,14)(7,18)(11,25,26,31)(12,29,30,20)(17,33)(19,23)(24,32,28,36)(34,35)Polycyclic presentation:
2^9|a^2=h,b^2=efg,c^2=fgh,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 298207);