Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30437 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×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×#3020 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1907 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)(6,11)(7,15)(8,20)(9,24)(12,26)(14,25)(16,29)(17,33)(21,27)(22,23)(31,35)(32,36);(5,6)(7,16)(8,21)(9,25)(10,14)(11,24)(12,19)(13,22)(15,26)(17,34)(18,29)(20,30)(23,27)(28,31)(32,33)(35,36);(5,7)(6,12)(8,17)(9,18)(10,15)(11,26)(13,28)(14,29)(16,25)(19,24)(20,35)(21,32)(22,33)(23,31)(27,36)(30,34);(5,8,10,23)(6,13,14,30)(7,17,15,31)(9,20,24,22)(11,27,25,21)(12,28,29,34)(16,32,26,36)(18,35,19,33)Polycyclic presentation:
2^9|a^2=h,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,g^c=gi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290366);