Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #111358 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #390 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | #1438 |
| prime radical | #60688 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
896 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 896 conjugacy classes of subgroups — too many to draw. Only its 146 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:
(2,5,6,7)(9,10)(11,19,21,15)(12,23,25,16)(13,18)(14,17)(20,37)(22,35,27,31)(24,40)(26,36,28,33)(29,34)(30,38)(32,39);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,24)(13,27)(14,21)(16,32)(17,19)(18,35)(20,26)(22,29)(23,40)(25,39)(28,38)(30,33)(31,34)(36,37);(1,2)(3,7)(4,6)(5,8)(9,12)(10,16)(11,20)(13,28)(14,25)(15,30)(17,23)(18,36)(19,37)(21,38)(22,39)(24,27)(26,29)(31,40)(32,35)(33,34)Polycyclic presentation:
2^9|a^2=g,d^2=i,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,d^b=di,e^b=ehi,f^b=fh,g^b=ghi,d^c=dhi,e^c=ei,f^c=fh,g^c=ghi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72237274, #72237275, #72239727, … |
| maximal normal subgroup of | #72237274, #72237275, #72239727, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 18554);