Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6631461 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2840 |
| commutator, Frattini, cosocle | C24 | C25 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
2071 conjugacy classes of subgroups; only the 488 normal subgroups are shown.
This group has 2071 conjugacy classes of subgroups — too many to draw. Only its 488 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 — 5 generators on 32 points:
(1,2)(3,11,14,7)(4,8)(5,18,20,9)(6,10)(12,21,15,26)(13,22)(16,23,19,25)(17,24)(27,30)(28,31)(29,32);(1,3)(2,7)(4,15)(5,13)(6,14)(8,21)(9,22)(10,11)(12,17)(16,29)(18,27)(19,28)(20,30)(23,31)(24,26)(25,32);(1,4)(2,8)(3,12,14,15)(5,19,20,16)(6,17)(7,21,11,26)(9,25,18,23)(10,24)(13,29)(22,32)(27,31)(28,30);(1,5)(2,9)(3,13)(4,16)(6,20)(7,22)(8,23)(10,18)(11,27)(12,28)(14,30)(15,29)(17,19)(21,31)(24,25)(26,32);(3,14)(7,11)(12,15)(13,30)(21,26)(22,27)(28,29)(31,32)Polycyclic presentation:
2^9|a^2=g,c^2=g,b^a=bf,c^a=cg,d^a=dh,f^a=fi,h^a=hi,c^b=ch,e^b=ei,g^b=gi,d^c=dfh,h^c=hi,g^d=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72210026, #72210037, #72235133, … |
| maximal normal subgroup of | #72210026, #72210037, #72235133, … |
| maximal quotient of | #72228236, #72228245, #72235173, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6538657);