Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #486147 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C22⋊C4 | C24 |
| prime radical | #12093 | C2 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
846 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 48 points:
(1,2)(3,9)(4,12)(5,7)(6,11)(8,14)(10,16)(13,15)(17,18)(19,27)(20,24)(21,25)(22,26)(23,30)(28,39)(29,40)(31,36)(32,37)(33,38)(34,42)(35,43)(41,47)(44,46)(45,48);(1,3)(2,6)(4,13)(5,10)(7,15)(8,16)(9,12)(11,14)(17,19)(18,23)(20,29)(21,30)(22,28)(24,35)(25,27)(26,34)(31,43)(32,41)(33,42)(36,40)(37,45)(38,39)(44,48)(46,47);(19,28)(21,33)(23,34)(27,39)(29,41)(31,44)(35,45)(40,47);(2,7)(3,10)(8,12)(11,13)(17,20,22,32)(18,24,26,37)(19,29,28,41)(21,31,33,44)(23,35,34,45)(25,36,38,46)(27,40,39,47)(30,43,42,48)Polycyclic presentation:
2^9|d^2=i,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,h^a=hi,c^b=ci,d^b=dg,e^b=eg,f^b=fhi,e^c=eh,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72284305, #72285696, #72285719, … |
| maximal normal subgroup of | #72284305, #72285696, #72285719, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 393344);