Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500807 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24260 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| prime radical | #33559 | C2 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1037 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 1037 conjugacy classes of subgroups — too many to draw. Only its 138 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:
(2,8)(4,17)(5,10)(6,20)(9,22)(11,27)(12,29)(13,31)(14,25)(18,23)(24,30)(28,32);(1,2,5,8,7,4,10,17)(3,12,14,29,16,28,25,32)(6,18,19,27,20,11,26,23)(9,24,22,21,13,30,31,15);(1,3)(2,9)(4,13)(5,14)(6,21)(7,16)(8,22)(10,25)(11,28)(12,18)(15,20)(17,31)(19,30)(23,32)(24,26)(27,29);(1,4,5,17,7,2,10,8)(3,13,14,31,16,9,25,22)(6,18,19,27,20,11,26,23)(12,30,29,15,28,24,32,21)Polycyclic presentation:
2^9|b^2=e,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=ghi,c^b=cg,f^b=fh,g^b=gi,g^c=gi,h^c=hi,g^f=gi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72236886, #72236900, #72237519, … |
| maximal normal subgroup of | #72236900, #72237519, #72237525, … |
| maximal quotient of | #72311838, #72311842, #72311847, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408003);