Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #458125 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #555 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2420 |
| prime radical | #23438 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1045 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,16,12,17,11,15)(7,18,9,13,21,29,20,30)(8,22,25,33,27,32,26,14)(19,34,35,36,24,28,23,31);(1,2,3,4)(5,7,11,20,12,21,10,9)(6,13,16,29,17,30,15,18)(8,23,26,24,27,35,25,19)(14,31,22,34,33,36,32,28);(2,4)(5,8)(6,14)(7,19)(9,23)(10,26)(11,25)(12,27)(13,28)(15,22)(16,32)(17,33)(18,31)(20,35)(21,24)(29,36)(30,34);(5,9,10,21,12,20,11,7)(6,13,16,29,17,30,15,18)(8,24,25,23,27,19,26,35)(14,31,22,34,33,36,32,28)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=f,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 373046);