Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #459444 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2749 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1120 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,12,16,11,18,10,17)(7,19,25,33,23,32,22,13)(8,14,21,31,27,35,26,34)(9,28,24,15,20,36,29,30);(1,2)(3,4)(5,7,11,23)(6,13)(8,24,27,29)(9,21,20,26)(10,22,12,25)(14,28)(15,31)(16,19)(17,32)(18,33)(30,34)(35,36);(2,4)(5,8,11,27)(6,14,18,35)(7,20,23,9)(10,21,12,26)(13,30,33,15)(16,34,17,31)(19,36,32,28)(22,29,25,24);(5,9)(6,15,18,30)(7,21)(8,25)(10,24)(11,20)(12,29)(13,31,33,34)(14,32,35,19)(16,28,17,36)(22,27)(23,26)Polycyclic presentation:
2^9|a^2=ef,b^2=f,c^2=i,d^2=fi,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,g^b=gi,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 366643);