Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23414 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#127 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2397 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1390 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,16,20,17)(8,22)(9,14)(10,26)(11,15)(12,19,28,21)(13,25)(18,31,33,30)(23,36)(24,29)(27,34,35,32);(2,4)(5,7)(6,12)(8,18)(9,21)(10,19)(11,20)(13,27)(14,17)(15,28)(16,26)(22,35)(23,34)(24,32)(25,33)(29,31)(30,36);(2,4)(5,6)(7,17)(8,13)(9,14)(10,26)(11,15)(12,19)(16,20)(18,31)(21,28)(22,25)(23,29)(24,36)(27,32)(30,33)(34,35);(5,8,9,23,11,25,10,24)(6,13,14,29,15,22,26,36)(7,18,21,34,20,33,19,32)(12,27,17,31,28,35,16,30)Polycyclic presentation:
2^9|a^2=h,d^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72226715, #72239887, #72250587, … |
| maximal normal subgroup of | #72226715, #72239887, #72250587, … |
| maximal quotient of | #72226711, #72226712, #72239888 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265863);