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 | #2535 | C4 |
| LCS | G ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×D4×Q8 ◁ G | |
| LECS | G ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1 | |
707 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 707 conjugacy classes of subgroups — too many to draw. Only its 88 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:
(1,2,6,7,5,3,4,8)(9,10)(11,20,23,35)(12,17,25,32)(13,18,26,16)(14,19)(15,21)(22,34)(24,31)(27,36,28,40)(29,37)(30,39)(33,38);(1,3,5,2)(4,8,6,7)(9,11,14,23)(10,15,19,29)(12,21,25,37)(13,22,26,38)(16,27,18,28)(17,20,32,35)(24,36,39,40)(30,33,31,34);(1,4,5,6)(2,7,3,8)(10,16)(13,26)(15,27)(17,30)(18,19)(20,33)(22,38)(24,39)(28,29)(31,32)(34,35)(36,40)Polycyclic presentation:
2^9|a^2=f,b^2=hi,c^2=h,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.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 38182);