Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #213818 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1615 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6)(7,15,18,11)(8,19,21,34)(9,13)(10,14)(12,20,27,36)(16,30)(17,28,22,25)(23,29)(24,31)(26,35)(32,33);(1,2)(3,4)(5,7)(6,11)(8,16)(9,22)(10,18)(12,26)(13,28)(14,15)(17,23)(19,33)(20,35)(21,32)(24,27)(25,29)(30,34)(31,36);(5,8,10,21)(6,12)(7,16,18,32)(9,20,23,36)(11,24)(13,19)(14,27)(15,26)(17,31,22,35)(25,33)(28,30)(29,34);(1,3,4,2)(6,13,14,29)(11,25,15,28)(12,19,27,34)(24,33,26,30)Polycyclic presentation:
2^9|a^2=h,c^2=gi,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^b=hi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72286176, #72286206, #72286224 |
| maximal normal subgroup of | #72286176, #72286206, #72286224 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293266);