Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23995 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2580 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1114 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,11,16,12,17,10,15)(7,13,21,9,19,27,20,26)(8,14,25,31,24,30,28,32)(18,33,22,36,34,29,35,23);(1,2)(3,4)(5,7)(6,13)(8,22,24,35)(9,16)(10,20)(11,21)(12,19)(14,23,30,36)(15,26)(17,27)(18,25,34,28)(29,31,33,32);(2,4)(5,8)(6,14)(7,18)(9,23)(10,28)(11,25)(12,24)(13,29)(15,32)(16,31)(17,30)(19,34)(20,35)(21,22)(26,36)(27,33);(5,9,11,27,12,26,10,13)(6,7,16,21,17,19,15,20)(8,23,25,33,24,36,28,29)(14,18,31,22,30,34,32,35)Polycyclic presentation:
2^9|a^2=h,b^2=e,d^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,f^b=fi,e^c=ei,g^c=gi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72307762 |
| maximal normal subgroup of | #72307762 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 283138);