Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #130114 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C2×C4⋊C4 | C22×C4 |
| Frattini, cosocle | C2×C4×Q8 | C23 |
| prime radical | #2828 | C4 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1 | |
539 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 539 conjugacy classes of subgroups — too many to draw. Only its 84 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:
(1,2,5,7,6,3,4,8)(9,10)(11,20,23,37)(12,17,25,34)(13,16)(14,19)(15,21)(18,27)(22,35,26,36)(24,33,28,32)(29,38)(30,40)(31,39);(1,3)(2,6)(4,8)(5,7)(9,11,14,23)(10,15,19,31)(12,21,25,39)(13,26,27,22)(16,29,18,30)(17,20,34,37)(24,40,28,38)(32,36,33,35);(1,4,6,5)(2,7,3,8)(10,16)(13,27)(15,29)(17,33)(18,19)(20,35)(22,26)(24,28)(30,31)(32,34)(36,37)(38,40)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=h,d^2=h,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,d^b=dh,e^b=eg,f^b=fgh,d^c=dgi,e^c=ei,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, 37616);