Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453905 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16211 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
814 conjugacy classes of subgroups; only the 160 normal subgroups are shown.
This group has 814 conjugacy classes of subgroups — too many to draw. Only its 160 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)(3,11)(4,15)(5,9)(6,10)(7,14)(8,18)(12,20)(13,21)(16,24)(17,22)(19,23)(25,27)(26,31)(28,30)(29,32);(1,3)(2,7,10,11)(4,16)(5,13,19,27)(6,14)(8,20,15,24)(9,21)(12,18)(17,26,30,29)(22,32)(23,25)(28,31);(1,4)(2,8,10,15)(3,12,14,16)(5,17)(6,18)(7,20)(9,22,23,28)(11,24)(13,26,27,29)(19,30)(21,31)(25,32);(1,5)(2,9)(3,13,14,27)(4,17)(6,19)(7,21,11,25)(8,22)(10,23)(12,26,16,29)(15,28)(18,30)(20,31,24,32)Polycyclic presentation:
2^9|b^2=efgh,c^2=eg,d^2=ef,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^b=ei,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72318171, #72318194, #72318514, … |
| maximal normal subgroup of | #72318171, #72318194, #72318514, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361101);