Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8528 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #379 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#379 |
| prime radical | #62759 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2364 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,6,5,3,7,8)(9,10,15,19,14,20,13,18)(11,16,21,24,17,12,22,23);(1,3)(2,5)(4,6)(7,8)(9,11)(10,16)(12,20)(13,21)(14,17)(15,22)(18,24)(19,23);(4,7)(6,8)(9,12)(10,17)(11,20)(13,23)(14,16)(15,24)(18,22)(19,21);(9,13,14,15)(10,18,20,19)(11,21,17,22)(12,23,16,24)Polycyclic presentation:
2^9|a^2=g,d^2=h,e^2=i,f^2=h,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,e^b=ei,f^b=fhi,g^b=ghi,e^c=ehi,f^c=fh,g^c=ghi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72240112, #72240115, #72240120, … |
| maximal normal subgroup of | #72240112, #72240115, #72240120, … |
| maximal quotient of | #72340581, #72340652 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 95092);