Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62243 |
| 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 | #61432 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C8×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1603 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,6)(4,11)(7,13)(8,12)(10,15)(14,16)(17,18,22,25,23,19,21,27)(20,26,29,32,30,24,28,31);(1,2,5,8)(3,7,9,14)(4,12,11,6)(10,16,15,13)(17,19,23,18)(20,24,30,26)(21,25,22,27)(28,32,29,31);(17,20,22,29,23,30,21,28)(18,24,25,31,19,26,27,32);(1,3)(2,7)(4,10)(5,9)(6,13)(8,14)(11,15)(12,16);(3,9)(7,14)(10,15)(13,16)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,f^2=h,g^2=i,b^a=bf,c^a=ci,f^a=fh,c^b=ci,f^b=fh,e^d=ehi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72338407, #72338443, #72338446 |
| maximal normal subgroup of | #72338407, #72338443, #72338446 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6253169);