Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #245063 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C26 | D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1680 conjugacy classes of subgroups; only the 176 normal subgroups are shown.
This group has 1680 conjugacy classes of subgroups — too many to draw. Only its 176 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:
(1,2,6,10)(3,11,14,22)(4,15,18,8)(5,9,19,24)(7,16,21,13)(12,25,27,32)(17,28,30,23)(20,29,31,26);(7,20)(11,25)(21,31)(22,32);(1,3,6,14)(2,7,10,21)(4,16,18,13)(5,12,19,27)(8,22,15,11)(9,20,24,31)(17,29,30,26)(23,32,28,25);(4,17)(8,23)(11,25)(13,26)(15,28)(16,29)(18,30)(22,32)Polycyclic presentation:
2^9|a^2=i,c^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gh,c^b=cg,e^c=eh,f^c=fi,f^d=fh56 conjugacy classes — view the full character table.
| maximal subgroup of | #72227596, #72227898 |
| maximal normal subgroup of | #72227596, #72227898 |
| maximal quotient of | #72209840, #72209853, #72209867, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 152270);