Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393937 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14715 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3967 conjugacy classes of subgroups; only the 296 normal subgroups are shown.
This group has 3967 conjugacy classes of subgroups — too many to draw. Only its 296 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:
(5,17)(8,22)(9,18)(11,24)(12,26)(14,19)(16,28)(25,32);(1,2)(3,11)(4,8)(5,18)(6,10)(7,14)(9,17)(12,25)(13,24)(15,22)(16,29)(19,21)(20,27)(23,28)(26,32)(30,31);(1,3)(2,7)(4,12)(5,19)(6,13)(8,20)(9,24)(10,21)(11,18)(14,17)(15,26)(16,30)(22,31)(23,32)(25,29)(27,28);(1,4)(2,8)(3,12)(5,16)(6,15)(7,20)(9,23)(10,22)(11,25)(13,26)(14,27)(17,28)(18,29)(19,30)(21,31)(24,32)Polycyclic presentation:
2^9|g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,g^b=gi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,e^d=ei,f^d=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72215906, #72215925, #72215953, … |
| maximal normal subgroup of | #72215906, #72215925, #72215953, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301133);