Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #141751 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1957 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #3169 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
316 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 316 conjugacy classes of subgroups — too many to draw. Only its 94 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,4,7,6,10,5,9)(3,11,13,16,12,15,14,8)(17,18)(19,27,30,40)(20,32)(21,24)(22,36)(23,29,37,33)(25,28)(26,35)(31,46,45,41)(34,47)(38,42,48,44)(39,43);(2,7,10,9)(3,12)(8,15,16,11)(13,14)(17,19,28,42,35,45,20,33,21,30,43,44,22,31,34,29)(18,23,32,46,36,48,25,40,24,37,47,41,26,38,39,27);(1,3,5,14,6,12,4,13)(2,8,7,11,10,16,9,15)(18,24)(23,37)(25,39)(26,36)(27,40)(32,47)(38,48)(41,46)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=ei,d^2=g,e^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,c^b=ch,d^b=di,f^b=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72225269, #72225276, #72287554, … |
| maximal normal subgroup of | #72225269, #72225276, #72287554, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 49034);