Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24408 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2227 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,17,20,12)(8,21)(9,16)(10,15)(11,14)(13,24)(18,34,35,29)(19,28,26,32)(22,33,36,27)(23,30)(25,31);(1,2,3,4)(5,7)(6,12)(8,22)(9,19)(10,26)(11,20)(13,29)(14,17)(15,28)(16,32)(18,25)(21,33)(23,36)(24,35)(27,30)(31,34);(2,4)(5,8)(6,13)(7,18)(9,25)(10,24)(11,23)(12,27)(14,31)(15,30)(16,21)(17,33)(19,36)(20,35)(22,26)(28,34)(29,32);(5,7)(6,12)(8,18)(9,26)(10,19)(11,20)(13,27)(14,17)(15,32)(16,28)(21,34)(22,25)(23,35)(24,36)(29,30)(31,33)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 273839);