Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419810 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2008 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1607 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)(7,12,18,28)(8,20,22,35)(9,15)(10,25)(11,14)(13,24,30,23)(16,29,19,26)(17,32)(21,34)(27,33)(31,36);(1,2)(3,4)(5,7)(6,12)(8,21)(9,18)(10,16)(11,19)(13,27)(14,26)(15,28)(17,22)(20,34)(23,33)(24,31)(25,29)(30,36)(32,35);(2,4)(7,16)(10,11)(12,26)(14,25)(17,31)(18,19)(20,35)(21,33)(23,24)(27,34)(28,29)(32,36);(5,8)(6,13,15,30)(7,17)(9,22)(10,23)(11,24)(12,27,28,36)(14,20,25,35)(16,31)(18,21)(19,33)(26,34,29,32)Polycyclic presentation:
2^9|a^2=f,d^2=ei,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^c=hi,f^d=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72252199, #72256284, #72256289 |
| maximal normal subgroup of | #72252199, #72256284, #72256289 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327022);