Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #485605 |
| 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 | C23×D4 | D4 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
843 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 843 conjugacy classes of subgroups — too many to draw. Only its 120 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) = 64 |
A minimal faithful permutation representation — 4 generators on 48 points:
(1,2,5,8)(3,9,10,16)(4,12,14,7)(6,13,15,11)(17,18)(19,27)(20,24)(21,25)(22,26)(23,29)(28,39)(30,40)(31,36)(32,37)(33,38)(34,42)(35,43)(41,47)(44,46)(45,48);(1,3)(2,6)(4,13)(5,10)(7,16)(8,15)(9,12)(11,14)(17,19,22,30)(18,23,26,35)(20,28,31,41)(21,29,33,43)(24,34,36,45)(25,27,38,40)(32,42,44,48)(37,39,46,47);(20,31)(21,33)(24,36)(28,41)(29,43)(34,45)(37,46)(39,47);(3,10)(6,15)(9,16)(11,13)(17,20,22,31)(18,24,26,36)(19,28,30,41)(21,32,33,44)(23,34,35,45)(25,37,38,46)(27,39,40,47)(29,42,43,48)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=i,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=hi,d^b=dg,e^b=eg,f^b=fhi,d^c=di,e^c=eh,f^e=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72228029, #72228051, #72283548, … |
| maximal normal subgroup of | #72228029, #72283548, #72284170, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 392885);