Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6361570 |
| 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 |
| prime radical | #61437 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1785 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,8)(3,6,4,7)(9,10,14,12)(11,17,18,24)(13,21,23,16)(15,22,20,19);(1,3,5,4)(2,6,8,7)(9,11)(10,15)(12,20)(13,22)(14,18)(16,24)(17,21)(19,23);(2,7)(3,4)(6,8);(2,8)(6,7)(9,12,14,10)(11,17,18,24)(13,16,23,21)(15,22,20,19);(11,18)(15,20)(17,24)(19,22)Polycyclic presentation:
2^9|a^2=hi,b^2=i,d^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fh,g^a=gi,c^b=ci,d^b=df,e^b=eh,f^b=fh,g^c=gi,f^d=fh65 conjugacy classes — view the full character table.
| maximal subgroup of | #72237138 |
| maximal normal subgroup of | #72237138 |
| maximal quotient of | #72279630, #72279848, #72309459, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6268803);