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×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1079 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)(7,16)(8,21)(9,10)(12,17)(13,27)(14,23)(19,20)(22,24)(26,30)(28,29)(31,35)(32,33);(5,6)(7,17)(8,13)(9,14)(10,23)(11,15)(12,20)(16,26)(18,33)(19,30)(21,27)(22,29)(24,28)(25,31)(32,34)(35,36);(5,7)(6,12)(8,18)(9,20)(10,16)(11,19)(13,25)(14,30)(15,26)(17,23)(21,35)(22,34)(24,31)(27,33)(28,32)(29,36);(5,8,10,24,11,22,9,21)(6,13,23,28,15,29,14,27)(7,18,16,31,19,34,20,35)(12,25,17,32,26,36,30,33)Polycyclic presentation:
2^9|a^2=h,d^2=efi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,g^a=gi,c^b=cg,d^b=de,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, 292031);