Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #481284 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#102 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#3584 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
3158 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)(6,11,13,25)(8,18)(9,21)(12,24,27,35)(14,19,22,26)(16,31)(17,30)(20,23)(28,32,33,36)(29,34);(5,6)(7,12)(8,19)(9,22)(10,13)(11,23)(14,21)(15,27)(16,32)(17,33)(18,25)(20,26)(24,34)(28,30)(29,36)(31,35);(2,4)(5,7)(6,12)(8,16)(9,17)(10,15)(11,24)(13,27)(14,28)(18,31)(19,32)(20,29)(21,30)(22,33)(23,34)(25,35)(26,36);(7,15)(12,27)(16,29)(17,30)(24,35)(28,33)(31,34)(32,36)Polycyclic presentation:
2^9|a^2=g,e^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=ehi,g^b=gi,h^b=hi,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72225127, #72249860, #72249882, … |
| maximal normal subgroup of | #72225127, #72249860, #72249882, … |
| maximal quotient of | #72225044, #72285982, #72287164, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400719);