Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13595 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#354 |
| commutator | C42 | C2×C42 |
| Frattini, cosocle | C2×#127 | C23 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C2×#127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
668 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 668 conjugacy classes of subgroups — too many to draw. Only its 130 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6)(7,16,19,26)(8,12)(9,14,23,13)(10,15)(11,18,25,34)(17,31,33,27)(20,35,36,24)(21,29,32,28)(22,30);(1,2,3,4)(5,7,9,19,10,18,23,34)(6,11,14,16,15,26,13,25)(8,20,21,36,22,17,32,33)(12,27,29,35,30,24,28,31);(5,8,10,22)(6,12,15,30)(7,17,18,20)(9,21,23,32)(11,24,26,27)(13,28,14,29)(16,31,25,35)(19,33,34,36)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=i,d^2=hi,f^2=i,g^2=i,b^a=bd,d^a=dg,f^a=fh,g^a=gh,h^a=hi,c^b=ci,d^b=di,e^b=egh,e^d=ehi,f^e=fi,g^e=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72286468, #72286477, #72286495, … |
| maximal normal subgroup of | #72286468, #72286477, #72286495, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59044);