Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #232771 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#127 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#392 |
| prime radical | #60940 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1495 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,6)(3,7)(4,8)(9,10)(11,18)(12,16)(13,21)(14,17)(15,19)(20,24)(22,23);(1,2)(3,5)(4,7)(6,8)(9,11)(10,15)(12,19)(13,22)(14,20)(16,24)(17,23)(18,21);(1,3,5,2)(4,6,8,7)(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,18,23,24);(11,20)(15,23)(18,24)(19,22)Polycyclic presentation:
2^9|c^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,d^b=dhi,e^b=eh,f^b=fh,e^c=ei,f^c=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72303270, #72303383, #72303461, … |
| maximal normal subgroup of | #72303270, #72303383, #72303461, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 140788);