Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #125092 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1530 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C2×#97 | C23 |
| prime radical | C2×#127 | C2×C4 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #1726 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
267 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 267 conjugacy classes of subgroups — too many to draw. Only its 66 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 — 3 generators on 48 points:
(1,2)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(17,18,20,27,22,26,33,41,23,28,34,42,21,24,35,37)(19,25,30,39,29,38,43,47,32,40,44,48,31,36,45,46);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(18,24)(19,29)(20,33)(21,22)(26,28)(27,42)(30,44)(31,32)(34,35)(36,38)(39,46)(47,48);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,19,21,31,23,32,22,29)(18,25,24,36,28,40,26,38)(20,30,35,45,34,44,33,43)(27,39,37,46,42,48,41,47)Polycyclic presentation:
2^9|a^2=e,c^2=f,d^2=g,e^2=h,f^2=i,h^2=i,b^a=bd,d^a=df,g^a=gi,c^b=ch,d^b=dg,e^b=efgi,f^b=fi,h^b=hi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 32288);