Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6367198 |
| 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 | #61986 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1531 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,7,12)(3,8,4,14)(5,11,6,10)(9,13,16,15)(18,24)(19,26)(21,22)(25,32)(27,28)(29,31);(1,3,7,4)(2,8,12,14)(5,15,6,13)(9,10,16,11)(17,18)(19,23)(20,28)(21,26)(22,24)(25,29)(27,31)(30,32);(1,4,7,3)(2,9,12,16)(5,15,6,13)(8,10,14,11)(17,19,23,18)(20,27,30,25)(21,26,22,24)(28,29,32,31);(17,20)(18,25)(19,27)(21,31)(22,29)(23,30)(24,32)(26,28);(1,5,7,6)(2,10,12,11)(3,13,4,15)(8,9,14,16)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,c^2=i,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,f^b=fh,g^c=gi,f^d=fh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72283574, #72283751 |
| maximal normal subgroup of | #72283574, #72283751 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6274688);