Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394127 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3586 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3854 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)(5,6,12,17)(7,18,23,31)(8,24,27,28)(9,26,14,29)(10,16,11,15)(13,22,32,21)(19,35,34,30)(20,25,33,36);(2,4)(5,7)(6,13)(8,25)(9,20)(10,22)(11,21)(12,23)(14,33)(15,31)(16,18)(17,32)(19,26)(24,30)(27,36)(28,35)(29,34);(5,8)(6,14)(7,19)(9,17)(10,29)(11,26)(12,27)(13,30)(15,28)(16,24)(18,20)(21,36)(22,25)(23,34)(31,33)(32,35);(5,9,12,14)(6,8,17,27)(7,20,23,33)(10,28,11,24)(13,25,32,36)(15,26,16,29)(18,34,31,19)(21,30,22,35)Polycyclic presentation:
2^9|a^2=i,d^2=i,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,g^a=gi,h^a=hi,c^b=cg,e^b=ei,g^b=gi,e^c=ei,f^c=fi,g^c=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301326);