Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61569 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #3186 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#431 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1576 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)(3,12)(4,11)(5,9)(6,10)(7,13)(8,14)(15,16)(17,18,22,25,23,19,21,26)(20,27,29,32,30,24,28,31);(1,3)(2,7)(4,13)(5,15)(6,14)(8,11)(9,12)(10,16)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,31)(29,32);(1,3)(2,8)(4,15)(5,13)(6,14)(7,11)(9,16)(10,12)(17,20,22,29,23,30,21,28)(18,24,25,31,19,27,26,32);(1,4,6,5)(2,9,10,11)(3,13,14,15)(7,12,16,8)(17,21,23,22)(18,25,19,26)(20,28,30,29)(24,31,27,32);(2,10)(7,16)(8,12)(9,11)Polycyclic presentation:
2^9|a^2=g,c^2=g,d^2=h,f^2=h,g^2=i,b^a=bf,c^a=ch,d^a=dh,e^a=ehi,f^a=fh,d^b=di,f^b=fh,d^c=dhi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6252711);