Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11677 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×#102 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
384 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,7)(4,5)(6,8)(9,10,12,15,13,16,11,14)(18,24,27,32)(19,29)(20,31)(21,22)(25,26,30,28);(2,5)(6,7)(10,14,16,15)(17,18,19,30,21,26,29,24,23,27,31,25,22,28,20,32);(1,3,4,8)(2,6,5,7)(17,19,21,29,23,31,22,20)(18,25,26,32,27,30,28,24)Polycyclic presentation:
2^9|a^2=f,b^2=def,c^2=e,d^2=g,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,c^b=ci,d^b=di,f^b=fi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72343318, #72343320, #72343323, … |
| maximal normal subgroup of | #72343318, #72343320, #72343323, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 49085);