Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #135932 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #442 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×#322 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
232 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 232 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,6)(4,5)(7,8)(9,10,14,11,13,15,12,16)(17,18,19,24,22,27,20,26)(21,30,28,25,31,23,29,32);(2,5)(6,8)(9,11,14,15,13,16,12,10)(17,19,22,20)(21,29,31,28)(23,30)(25,32);(1,3,4,7)(2,6,5,8)(9,12,13,14)(10,11,15,16)(18,23)(19,20)(21,29)(24,25)(26,32)(27,30)(28,31)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=dg,d^2=i,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gi,c^b=ch,e^b=ei,d^c=di,e^c=egi,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72320433, #72320473, #72320588, … |
| maximal normal subgroup of | #72320433, #72320473, #72320588, … |
| maximal quotient of | #72320703, C2×#140099 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47295);