Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #346224 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2760 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | #2403 | C22 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
647 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 647 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,6,10,13)(4,11,12,7)(5,8)(9,15)(14,16)(17,18)(19,27,28,23)(20,29)(21,30,32,25)(22,26)(24,31);(6,13)(9,16)(17,19)(18,23)(20,30)(21,24)(22,28)(25,31)(26,27)(29,32);(1,3)(2,6,8,13)(4,9)(5,10)(7,14,11,15)(12,16)(17,20)(18,24,26,29)(19,25,28,30)(21,27)(22,31)(23,32);(1,4)(2,7)(3,9)(5,12)(6,14)(8,11)(10,16)(13,15)(17,21,22,32)(18,25,26,30)(19,24,28,29)(20,27,31,23)Polycyclic presentation:
2^9|a^2=fg,c^2=ef,d^2=h,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fhi,g^a=ghi,c^b=cg,d^b=de,f^b=fh,e^c=ei,f^c=fi,g^c=gh,g^d=gi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72283465, #72283535, #72283625, … |
| maximal normal subgroup of | #72283465, #72283535, #72283625, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 253428);