Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476109 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23439 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1282 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)(5,6,10,13)(7,17,20,27)(8,22,25,32)(9,16,11,15)(12,21,29,18)(14,24,30,26)(19,28,34,36)(23,35,33,31);(1,2)(3,4)(5,7)(6,12)(8,23)(9,21)(10,20)(11,18)(13,29)(14,31)(15,27)(16,17)(19,26)(22,28)(24,34)(25,33)(30,35)(32,36);(2,4)(6,13)(7,18)(8,24)(9,11)(12,27)(14,22)(17,29)(20,21)(23,33)(25,26)(28,36)(30,32);(2,4)(5,8)(6,14)(7,19)(9,26)(10,25)(11,24)(12,28)(13,30)(15,22)(16,32)(17,31)(18,23)(20,34)(21,33)(27,35)(29,36)Polycyclic presentation:
2^9|a^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=def,e^b=ei,g^b=gi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei,h^d=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 383306);