Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417690 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2669 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3018 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2344 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)(5,6,12,18)(7,19,21,31)(8,14,25,34)(9,26,27,32)(10,17,11,16)(13,22,30,20)(15,23,35,28)(24,36,29,33);(5,7)(6,13)(8,23)(9,24)(10,22)(11,20)(12,21)(14,32)(15,33)(16,31)(17,19)(18,30)(25,28)(26,34)(27,29)(35,36);(2,4)(5,8)(6,14)(7,9)(10,24)(11,29)(12,25)(13,15)(16,33)(17,36)(18,34)(19,32)(20,23)(21,27)(22,28)(26,31)(30,35);(5,9)(6,15)(7,8)(10,28)(11,23)(12,27)(13,14)(16,26)(17,32)(18,35)(19,36)(20,29)(21,25)(22,24)(30,34)(31,33)Polycyclic presentation:
2^9|a^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 324887);