Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511842 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #3027 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| socle | C2 | #32976 |
| prime radical | #3483 | C22 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
753 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 753 conjugacy classes of subgroups — too many to draw. Only its 150 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:
(1,2,5,23,8,14,22,11)(3,15,18,9,16,29,30,27)(4,13,21,12,7,24,6,10)(17,28,31,26,20,25,19,32);(1,3,5,18,8,16,22,30)(2,9,11,15,14,27,23,29)(4,17,21,31,7,20,6,19)(10,28,24,32,12,25,13,26);(2,10)(3,16)(4,7)(6,21)(9,25)(11,24)(12,14)(13,23)(15,26)(18,30)(27,28)(29,32);(1,4)(3,17)(5,21)(6,22)(7,8)(10,12)(13,24)(16,20)(18,31)(19,30)(25,28)(26,32)Polycyclic presentation:
2^9|a^2=eh,b^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=ci,d^c=di,f^c=fi,g^c=ghi,f^d=fh,g^d=gi,g^f=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72231593, #72231703, #72231714, … |
| maximal normal subgroup of | #72231593, #72231703, #72231714, … |
| maximal quotient of | #72250465, #72279633, C2×#512045, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419241);