Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152191 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #12759 |
| commutator | #127 | C22×C4 |
| Frattini, cosocle | C2×#127 | C23 |
| prime radical | #61110 | C2 |
| LCS | G ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×#127 ◁ G | |
| LECS | G ▷ C2×#127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
858 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 858 conjugacy classes of subgroups — too many to draw. Only its 72 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 — 3 generators on 32 points:
(1,2)(3,12,15,7)(4,8)(5,10,20,9)(6,11)(13,28,30,32)(14,23,16,22)(17,26,19,25)(18,24)(21,29,27,31);(1,3)(2,7)(4,13)(5,14)(6,16)(8,21)(9,22)(10,12)(11,23)(15,20)(17,30)(18,31)(19,29)(24,28)(25,32)(26,27);(1,4)(2,8)(3,13)(5,19)(6,18)(7,21)(9,25)(10,26)(11,24)(12,27)(14,29)(15,30)(16,31)(17,20)(22,32)(23,28)Polycyclic presentation:
2^9|a^2=f,d^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,d^b=dhi,e^b=eghi,f^b=fgh,g^b=gi,d^c=dgi,g^c=gi,h^c=hi,e^d=ehi,f^d=fhi,g^e=gi,g^f=gi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72213671, #72216601, #72216603, … |
| maximal normal subgroup of | #72213671, #72216601, #72216603, … |
| maximal quotient of | #72231258, #72231260, #72231261, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59387);