Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #512663 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2127 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | #127 | C24 |
| prime radical | #60808 | C2 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C23×C4 ◁ #3449 ◁ G | |
| LECS | G ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
843 conjugacy classes of subgroups; only the 184 normal subgroups are shown.
This group has 843 conjugacy classes of subgroups — too many to draw. Only its 184 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) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,11)(3,9,15,23)(4,10,17,25)(6,8,18,21)(7,12,19,26)(13,24,27,32)(14,22,28,31)(16,20,29,30);(2,8)(3,13)(4,6)(5,18)(9,20)(10,12)(15,28)(17,19)(21,25)(22,24)(27,29)(30,31);(1,3)(2,9)(4,14)(5,15)(6,16)(7,13)(8,20)(10,22)(11,23)(12,24)(17,28)(18,29)(19,27)(21,30)(25,31)(26,32);(1,4,7,6)(2,10,12,8)(3,14,13,16)(5,17,19,18)(9,22,24,20)(11,25,26,21)(15,28,27,29)(23,31,32,30)Polycyclic presentation:
2^9|a^2=f,d^2=i,e^2=hi,g^2=i,b^a=be,e^a=eg,g^a=gh,h^a=hi,c^b=ch,d^b=di,e^b=ehi,f^b=fgh,g^b=gi,e^c=ei,f^e=fhi,g^f=gi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72284440, #72286866, #72287234, … |
| maximal normal subgroup of | #72284440, #72286866, #72287234, … |
| maximal quotient of | #72286495, C2×#512664, #72292961, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419860);