Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #237997 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#124 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#394 |
| prime radical | #60688 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
942 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)(3,7)(4,6)(5,8)(9,10)(11,18)(12,20)(13,21)(14,16)(15,19)(17,23)(22,24);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,17)(13,22)(14,20)(16,24)(18,21)(19,23);(2,6)(3,4)(7,8)(9,12,13,15,14,18,23,24)(10,11,17,19,16,20,21,22);(1,3,5,4)(2,7,8,6)(10,16)(12,18)(15,24)(17,21)Polycyclic presentation:
2^9|c^2=e,d^2=i,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,e^b=eh,f^b=fhi,g^b=gi,d^c=dhi,f^c=fhi,g^c=ghi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72264932, #72265113, #72265117, … |
| maximal normal subgroup of | #72264932, #72265113, #72265117, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 146482);