Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #213752 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C2×#534 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1602 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)(7,15,18,11)(8,19,21,12)(9,13)(10,14)(16,25)(17,24,31,27)(20,32,33,26)(22,28)(23,30)(29,34)(35,36);(2,4)(5,7)(6,11)(8,16)(9,17)(10,18)(12,25)(13,27)(14,15)(19,23)(20,29)(21,30)(22,31)(24,28)(26,34)(32,35)(33,36);(1,2,3,4)(5,8)(6,12)(7,16)(9,20)(10,21)(11,23)(13,26)(14,19)(15,25)(17,29)(18,30)(22,33)(24,34)(27,35)(28,32)(31,36);(5,9,10,22)(6,13,14,28)(7,17,18,31)(8,20,21,33)(11,24,15,27)(12,26,19,32)(16,29,30,36)(23,34,25,35)Polycyclic presentation:
2^9|a^2=h,c^2=eg,d^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^b=hi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72284265, #72284354, #72284430, … |
| maximal normal subgroup of | #72284265, #72284354, #72284430, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293065);