Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #130074 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C2×C4⋊C4 | C22×C4 |
| Frattini, cosocle | C2×C4×D4 | C23 |
| prime radical | #46236 | C2 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1 | |
700 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 700 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,4,6,5,8,3,7)(9,10)(11,21,24,36)(12,18,13,26)(14,19)(15,20)(16,22)(17,25,35,28)(23,31,29,40)(27,33)(30,34)(32,38)(37,39);(2,6)(3,4)(7,8)(9,11)(10,16)(12,22)(13,27)(14,29)(15,24)(17,31)(18,36)(19,32)(20,33)(21,26)(23,30)(25,39)(28,38)(34,37)(35,40);(9,12,15,13)(10,17,20,35)(11,22,24,27)(14,28,30,25)(16,31,33,40)(18,19,26,34)(21,37,36,32)(23,39,29,38)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=h,e^2=i,f^2=hi,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dh,e^b=egi,f^b=fgh,d^c=dg,e^c=ei,f^c=fi,e^d=ei,f^d=fi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 37415);