Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23472 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#124 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2398 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1354 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,10,16,12,17,11,15)(7,18,22,31,23,30,21,13)(8,24,9,14,27,32,26,33)(19,28,25,29,20,34,35,36);(5,7)(6,13)(8,25)(9,20)(10,22)(11,21)(12,23)(14,29)(15,30)(16,18)(17,31)(19,26)(24,28)(27,35)(32,34)(33,36);(2,4)(5,8,10,9,12,27,11,26)(6,14,15,24,17,33,16,32)(7,19,21,35,23,20,22,25)(13,28,18,29,31,34,30,36);(5,9,10,27,12,26,11,8)(6,14,16,32,17,33,15,24)(7,20,22,35,23,19,21,25)(13,29,18,34,31,36,30,28)Polycyclic presentation:
2^9|a^2=h,c^2=e,d^2=h,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,c^b=cg,e^b=ei,g^b=gi,d^c=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72250083, #72256876, #72256910, … |
| maximal normal subgroup of | #72250083, #72256876, #72256910, … |
| maximal quotient of | #72308396, #72343067 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266305);