Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #130114 |
| 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 | #2543 | C22 |
| 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 | |
684 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 684 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)(11,20)(12,17,24,15)(13,27)(14,19)(16,22)(18,30)(21,38,25,37)(23,36)(26,32,29,35)(28,34)(31,40,33,39);(2,6)(3,4)(7,8)(10,15)(12,24)(16,31)(17,19)(18,35)(20,37)(21,25)(26,29)(27,32)(33,34)(36,38)(39,40);(9,11,14,23)(10,16,19,34)(12,25,24,21)(13,28,30,22)(15,31,17,33)(18,36,27,20)(26,39,29,40)(32,37,35,38)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,e^a=ei,g^a=gi,d^b=dh,e^b=egi,f^b=fgh,d^c=dg,e^c=ei,e^d=ei,f^d=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72280964, #72281098, #72281110, … |
| maximal normal subgroup of | #72280964, #72281098, #72281110, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 37477);