Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30441 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1063 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(6,12)(7,16)(9,10)(14,29)(15,25)(17,26)(18,31)(19,20)(21,30)(22,23)(27,33)(34,35);(5,6)(7,17)(8,21)(9,25)(10,12)(11,15)(13,19)(14,22)(16,28)(18,33)(20,26)(23,30)(24,29)(27,35)(31,36)(32,34);(5,7,11,20)(6,13,15,28)(8,18,24,35)(9,16,10,19)(12,26,25,17)(14,27,30,33)(21,36,29,32)(22,31,23,34);(5,8)(6,14,15,30)(7,18)(9,22)(10,23)(11,24)(12,21,25,29)(13,27,28,33)(16,31)(17,32,26,36)(19,34)(20,35)Polycyclic presentation:
2^9|a^2=h,c^2=i,d^2=ef,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,g^b=gi,e^c=ei,f^c=fi,g^c=gi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 292060);