Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #113468 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1438 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | #60799 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
1107 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,4,6,5,3,7,8)(9,10,15,18)(11,12)(13,19,14,20)(16,22)(17,24)(21,23);(1,3)(2,5)(4,6)(7,8)(9,11)(10,16)(12,18)(13,17)(14,21)(15,22)(19,23)(20,24);(4,7)(6,8)(9,12)(10,17)(11,20)(13,24)(14,23)(15,16)(18,21)(19,22)Polycyclic presentation:
2^9|a^2=g,d^2=i,e^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,d^b=di,e^b=ehi,f^b=fh,g^b=ghi,d^c=dhi,e^c=eh,f^c=fh,g^c=ghi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72214356, #72214520, #72214564 |
| maximal normal subgroup of | #72214356, #72214520, #72214564 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 20667);