Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2081 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #397 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#397 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1614 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)(7,12,18,29)(8,20,23,32)(9,14)(10,24)(11,15)(13,22,31,25)(16,33)(17,27,19,28)(21,36)(26,34)(30,35);(1,2)(3,4)(5,7,9,17,11,18,10,19)(6,12,24,28,15,29,14,27)(8,21,22,16,23,34,25,35)(13,30,32,26,31,33,20,36);(2,4)(5,8,9,22,11,23,10,25)(6,13,24,32,15,31,14,20)(7,16,17,34,18,35,19,21)(12,26,28,33,29,36,27,30);(5,9,11,10)(6,14,15,24)(7,17,18,19)(8,22,23,25)(12,27,29,28)(13,20,31,32)(16,34,35,21)(26,30,36,33)Polycyclic presentation:
2^9|a^2=h,b^2=ef,c^2=ef,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72237172, #72237184, #72237189, … |
| maximal normal subgroup of | #72237172, #72237184, #72237189, … |
| maximal quotient of | #72239003 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265809);