Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61106 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3067 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2514 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,4,3)(5,6,10,16,11,14,9,15)(7,17,20,12,21,28,19,27)(8,13,24,32,25,30,23,31)(18,29,33,26,34,36,22,35);(1,3,4,2)(5,7)(6,12)(8,22)(9,20)(10,19)(11,21)(13,29)(14,27)(15,28)(16,17)(18,23)(24,34)(25,33)(26,31)(30,36)(32,35);(5,7)(6,12)(8,18)(9,19)(10,20)(11,21)(13,26)(14,27)(15,17)(16,28)(22,23)(24,33)(25,34)(29,31)(30,35)(32,36);(5,8,9,23,11,25,10,24)(6,13,15,31,14,30,16,32)(7,18,19,22,21,34,20,33)(12,26,17,29,27,35,28,36);(6,14)(12,27)(13,30)(15,16)(17,28)(26,35)(29,36)(31,32)Polycyclic presentation:
2^9|a^2=h,b^2=fh,d^2=f,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,d^b=df,f^b=fi,h^b=hi,g^c=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72279230, #72285705, #72285725, … |
| maximal normal subgroup of | #72279230, #72285705, #72285725, … |
| maximal quotient of | #72279482, #72309530, #72321848 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6289202);