Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454002 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2010 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1190 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6)(7,17)(8,22)(9,14)(10,15)(11,16)(12,19)(13,24)(18,34)(20,29)(21,28)(23,33)(25,32)(26,31)(27,36)(30,35);(1,2,4,3)(5,7,11,21)(6,12,16,29)(8,23,26,35)(9,20,10,19)(13,30,32,33)(14,17,15,28)(18,25,36,24)(22,34,31,27);(2,3)(5,8)(6,13)(7,18)(9,24)(10,25)(11,26)(12,27)(14,31)(15,22)(16,32)(17,33)(19,23)(20,35)(21,36)(28,30)(29,34);(1,3,4,2)(6,14)(8,24)(9,10)(12,17)(15,16)(18,23)(19,20)(22,31)(25,26)(27,34)(28,29)(35,36)Polycyclic presentation:
2^9|b^2=efi,d^2=ef,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,e^d=ei,f^d=fi,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 366517);