Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11234 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #391 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1445 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1594 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6,10,16,11,14,9,15)(7,17,20,32,19,12,25,27)(8,13,24,30,22,31,23,29)(18,26,21,28,34,33,35,36);(5,7)(6,12)(8,21,22,35)(9,20)(10,25)(11,19)(13,28,31,36)(14,17)(15,27)(16,32)(18,23,34,24)(26,29,33,30);(1,2,3,4)(5,8)(6,13)(7,18,19,34)(9,23)(10,24)(11,22)(12,26,17,33)(14,31)(15,29)(16,30)(20,21,25,35)(27,28,32,36)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,c^b=cf,e^b=ei,g^b=gi,d^c=di,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72226702, #72252097, #72252103, … |
| maximal normal subgroup of | #72226702, #72252097, #72252103, … |
| maximal quotient of | #72226696 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31113);