Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #131812 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1786 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#3287 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
858 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 858 conjugacy classes of subgroups — too many to draw. Only its 116 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) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,5,7,4,3,8,6)(9,10)(11,21,24,39)(12,16)(13,30)(14,31)(15,20)(17,23,36,27)(18,26)(19,28)(22,38,40,37)(25,33,32,35)(29,34);(5,8)(6,7)(10,16)(12,26)(13,29)(14,28)(17,33)(18,19)(20,34)(21,38)(22,23)(25,32)(27,40)(30,31)(35,36)(37,39);(1,3)(2,4)(5,6)(7,8)(9,11,15,24)(10,17,20,36)(12,27,29,23)(13,22,26,40)(14,32,28,25)(16,33,34,35)(18,21,30,39)(19,38,31,37)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=hi,e^2=g,f^2=ghi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,h^a=hi,d^b=dh,e^b=egi,f^b=fghi,g^b=gi,h^b=hi,d^c=dgi,e^c=egi,f^c=fi,g^c=gi,h^c=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72309441, #72309457, #72309461, … |
| maximal normal subgroup of | #72309441, #72309457, #72309461, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 39063);