Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #401009 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2018 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1469 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)(7,17)(8,22)(9,15)(10,13)(11,16)(12,20)(14,23)(18,28)(19,26)(21,27)(24,30)(25,29)(31,32)(33,35)(34,36);(2,4)(5,7)(6,12)(8,19)(9,18)(10,20)(11,21)(13,27)(14,26)(15,17)(16,28)(22,31)(23,33)(24,32)(25,34)(29,35)(30,36);(6,13)(7,18)(9,10)(14,29)(15,16)(17,27)(19,32)(20,21)(22,30)(23,24)(31,35)(33,34);(5,8)(6,14,16,30)(7,19)(9,24)(10,23)(11,25)(12,26,28,36)(13,29,15,22)(17,31,27,35)(18,32)(20,33)(21,34)Polycyclic presentation:
2^9|d^2=gh,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,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 321087);