Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #185475 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C22×#550 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
3610 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,6,5,7,3,8)(9,10,12,16,13,14,20,22)(11,17,18,24,19,15,21,23);(2,6)(3,4)(7,8)(10,14)(12,20)(15,17)(18,21);(9,11,13,19)(10,15,14,17)(12,21,20,18)(16,24,22,23);(2,7)(6,8)Polycyclic presentation:
2^9|a^2=g,c^2=h,e^2=i,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,e^b=ei,g^b=ghi,g^c=gh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72293026, #72302348, #72302452, … |
| maximal normal subgroup of | #72293026, #72302348, #72302452, … |
| maximal quotient of | #72293109 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 92953);