Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #130482 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
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 | #2397 | C4 |
| LCS | G ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#554 ◁ G | |
| LECS | G ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1 | |
681 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 681 conjugacy classes of subgroups — too many to draw. Only its 82 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:
(2,6,7,8)(9,10,14,20,15,18,29,34)(11,19,22,36,13,16,27,30)(12,23,24,38,25,35,28,39)(17,31,33,26,32,37,40,21);(1,2)(3,6)(4,8)(5,7)(9,11,15,13)(10,16,18,19)(12,21,25,26)(14,22,29,27)(17,23,32,35)(20,36,34,30)(24,31,28,37)(33,39,40,38);(1,3,5,4)(2,6,7,8)(9,12,15,25)(10,17,18,32)(11,21,13,26)(14,28,29,24)(16,23,19,35)(20,33,34,40)(22,37,27,31)(30,38,36,39)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=hi,d^2=h,e^2=i,f^2=i,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 | #72292959, #72292967, #72293121, … |
| maximal normal subgroup of | #72292959, #72292967, #72293121, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 37745);