Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418504 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1865 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(6,12)(7,15)(8,14)(9,10)(13,20)(17,18);(1,2)(3,4)(5,6)(7,13)(8,9)(10,12)(11,14)(15,19)(16,17)(18,20);(2,4)(5,7)(6,13)(8,16)(9,17)(10,15)(11,18)(12,19)(14,20);(5,8,10,14,11,12,9,6)(7,16,15,20,18,19,17,13)Polycyclic presentation:
2^9|d^2=ei,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,h^b=hi,f^c=fi,h^c=hi,f^d=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72319099, #72319359, #72319417, … |
| maximal normal subgroup of | #72319099, #72319359, #72319417, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325701);