Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #108372 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22⋊#95 |
| commutator | C42 | C2×C42 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
954 conjugacy classes of subgroups; only the 186 normal subgroups are shown.
This group has 954 conjugacy classes of subgroups — too many to draw. Only its 186 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)(3,7)(6,8)(9,10)(11,15,20,31)(12,22)(13,25)(14,17)(16,24)(18,28)(19,35,37,39)(21,33,26,30)(23,34)(27,32)(29,38,40,36);(1,2)(3,8)(4,6)(5,7)(9,11)(10,15,17,31)(12,19)(13,26)(14,20)(16,29,32,40)(18,33,25,30)(21,28)(22,39,34,35)(23,37)(24,38)(27,36);(3,7)(5,8)(9,12)(10,16)(11,19)(13,27)(14,23)(15,29)(17,32)(18,34)(20,37)(21,38)(22,25)(24,28)(26,36)(30,39)(31,40)(33,35)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=hi,e^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gi,d^b=dh,e^b=eh,f^b=fi,d^c=dh,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72286193, #72286216, #72286226, … |
| maximal normal subgroup of | #72286193, #72286216, #72286226, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 15568);