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 | #60946 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#431 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1642 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)(3,11)(4,7)(5,10)(6,9)(8,13)(12,15)(14,16)(17,18,21,24,22,19,26,25)(20,27,28,32,29,23,30,31);(1,3)(2,8)(4,12)(5,13)(6,15)(7,14)(9,11)(10,16)(17,19)(18,22)(20,27)(21,25)(23,29)(24,26)(28,32)(30,31);(1,4,7,2)(3,12,14,8)(5,10,6,9)(11,13,16,15)(17,20,26,30,22,29,21,28)(18,23,25,32,19,27,24,31);(2,4)(8,12)(9,10)(11,16)(18,19)(23,27)(24,25)(31,32);(1,5,7,6)(2,9,4,10)(3,13,14,15)(8,11,12,16)(17,21,22,26)(18,24,19,25)(20,28,29,30)(23,31,27,32)Polycyclic presentation:
2^9|a^2=g,c^2=gh,e^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, 6252713);