Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #137655 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1893 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C2×#3067 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
1014 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,10,15,12,8,11,24)(7,16,20,32,21,33,19,31)(9,25,27,23,29,36,28,14)(13,26,30,35,22,34,17,18);(5,7)(6,13)(8,22)(9,26)(10,20)(11,19)(12,21)(14,31)(15,30)(16,25)(17,24)(18,28)(23,32)(27,35)(29,34)(33,36);(2,4)(5,8)(6,12)(7,17)(9,14)(10,15)(11,24)(13,19)(16,26)(18,32)(20,22)(21,30)(23,29)(25,28)(27,36)(31,35)(33,34)Polycyclic presentation:
2^9|a^2=g,d^2=hi,e^2=i,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,c^b=cf,d^b=dh,e^b=ei,f^b=fi,h^b=hi,d^c=di,e^c=ei,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72239802, #72239807, #72239819, … |
| maximal normal subgroup of | #72239802, #72239807, #72239819, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54902);