Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454681 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2622 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3008 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1877 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,11,16)(7,17,21,28)(8,22)(9,13,10,15)(12,18,26,20)(14,23)(19,27)(24,29)(25,30)(31,32)(33,35)(34,36);(1,2)(3,4)(5,7)(6,12)(8,19)(9,18)(10,20)(11,21)(13,17)(14,27)(15,28)(16,26)(22,31)(23,34)(24,32)(25,33)(29,35)(30,36);(2,4)(6,13)(7,18)(8,23)(9,10)(12,26)(14,29)(15,16)(19,33)(20,21)(24,25)(27,31)(35,36);(5,8)(6,14)(7,19)(9,24)(10,23)(11,25)(12,27)(13,22)(15,30)(16,29)(17,31)(18,32)(20,34)(21,33)(26,35)(28,36)Polycyclic presentation:
2^9|a^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,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,h^c=hi,e^d=ei,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, 361877);