Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30437 |
| 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 | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1135 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)(7,16)(8,21)(9,12)(10,15)(11,18)(13,24)(14,25)(17,30)(19,28)(20,26)(22,27)(23,31)(29,35)(32,36)(33,34);(5,7)(6,11)(8,22)(9,19)(10,20)(12,16)(13,30)(14,26)(15,28)(17,23)(18,25)(21,34)(24,32)(27,31)(29,36)(33,35);(6,12)(9,25)(11,26)(13,29)(14,15)(16,28)(18,19)(21,31)(22,32)(23,35)(27,34)(30,36);(5,8,10,24)(6,13,15,21)(7,17)(9,23,25,35)(11,27)(12,29,14,31)(16,30)(18,22)(19,32)(20,33)(26,34)(28,36)Polycyclic presentation:
2^9|a^2=h,d^2=ef,e^2=i,f^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,e^c=ei,f^c=fi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 292056);