Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #130482 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C2×C4×Q8 | C23 |
| prime radical | #45387 | C2 |
| LCS | G ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#550 ◁ G | |
| LECS | G ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1 | |
911 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 911 conjugacy classes of subgroups — too many to draw. Only its 96 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)(11,21,14,19)(12,24)(13,18,28,34)(15,20)(16,23)(17,22,33,29)(25,30,38,40)(26,35)(27,31)(32,39)(36,37);(2,6)(3,4)(7,8)(9,11,15,14)(10,16,20,31)(12,22,26,29)(13,23,28,27)(17,30,33,40)(18,21,34,19)(24,37,35,32)(25,36,38,39);(9,12,15,26)(10,17,20,33)(11,22,14,29)(13,25,28,38)(16,30,31,40)(18,32,34,37)(19,35,21,24)(23,36,27,39)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=h,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,f^b=fgh,e^c=ei,f^c=fgi,f^d=fi,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72292982, #72292987, #72293175, … |
| maximal normal subgroup of | #72292982, #72292987, #72293175, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 37947);