Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6362950 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1807 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(9,10,14,12)(11,17,20,24)(13,22,23,16)(15,19,18,21);(9,11)(10,15)(12,18)(13,19)(14,20)(16,24)(17,22)(21,23);(2,5)(3,8)(6,7)(9,12,14,10)(11,18,20,15)(13,22,23,16)(17,21,24,19);(11,19)(13,23)(15,24)(16,22)(17,18)(20,21);(2,6)(5,7)(11,20)(15,18)(17,24)(19,21)Polycyclic presentation:
2^9|a^2=h,c^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fh,g^a=gi,d^b=df,e^b=eh,f^b=fh,g^c=gi,f^d=fh62 conjugacy classes — view the full character table.
| maximal subgroup of | #72237138 |
| maximal normal subgroup of | #72237138 |
| maximal quotient of | #72279250, #72280022, #72280057, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6270151);