Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2008 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #392 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1439 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1590 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)(5,6,10,14,11,15,9,26)(7,16,20,28,19,32,18,12)(8,21,23,30,25,29,24,13)(17,33,34,36,22,27,35,31);(2,4)(5,7)(6,12,15,28)(8,22,25,17)(9,18)(10,20)(11,19)(13,27)(14,16,26,32)(21,31)(23,35,24,34)(29,36)(30,33);(5,8,9,24,11,25,10,23)(6,13,14,21,15,30,26,29)(7,17,18,35,19,22,20,34)(12,27,16,31,28,33,32,36)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,d^2=i,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,h^a=hi,c^b=cf,f^b=fi,d^c=di,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72237171, #72237183, #72237189, … |
| maximal normal subgroup of | #72237171, #72237183, #72237189, … |
| maximal quotient of | #72238986 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31134);