Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C22×D4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C22 | C2×C8×D4 |
| prime radical | C23 | C8×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
418 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 48 points:
(1,2)(3,9)(4,12)(5,7)(6,10)(8,14)(11,16)(13,15)(17,18,22,27,21,28,37,45,23,25,39,42,38,26,43,44)(19,29,32,46,33,41,48,20,30,35,47,34,31,24,40,36);(1,3,5,11)(2,6,7,15)(4,13,14,10)(8,16,12,9)(17,19,23,30)(18,24,25,41)(20,27,36,42)(21,33,38,31)(22,32,39,47)(26,35,28,29)(34,45,46,44)(37,48,43,40);(2,7)(6,15)(8,12)(9,16)(17,20,22,35,21,34,37,24,23,36,39,29,38,46,43,41)(18,19,27,32,28,33,45,48,25,30,42,47,26,31,44,40);(18,25)(19,30)(26,28)(27,42)(31,33)(32,47)(40,48)(44,45)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,e^2=g,f^2=h,h^2=i,b^a=be,c^a=cg,d^a=di,e^a=eg,d^b=di,e^b=eg128 conjugacy classes — view the full character table.
| maximal subgroup of | #72321257, #72321277, #72321317, … |
| maximal normal subgroup of | #72321257, #72321277, #72321317, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390769);