Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417579 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23399 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1720 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)(10,16)(11,15)(12,20)(13,24)(18,33)(19,36)(21,28)(23,32)(25,31)(26,30)(27,35)(29,34);(1,2)(3,4)(5,7)(6,12)(8,18)(9,20)(10,25)(11,21)(13,29)(14,17)(15,31)(16,28)(19,23)(22,36)(24,35)(26,34)(27,30)(32,33);(5,8)(6,13)(7,18)(9,23)(10,26)(11,24)(12,27)(14,30)(15,32)(16,22)(17,29)(19,20)(21,35)(25,34)(28,33)(31,36);(1,3,4,2)(5,8)(6,13)(7,19)(9,23)(10,24)(11,26)(12,27)(14,30)(15,22)(16,32)(17,29)(18,20)(21,35)(25,34)(28,36)(31,33)Polycyclic presentation:
2^9|d^2=ei,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,f^c=fi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325601);