Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13603 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#354 |
| commutator | C42 | C2×C42 |
| Frattini, cosocle | C2×#124 | 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×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
660 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 660 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)(7,19,21,32)(8,14)(9,27)(10,16,11,17)(12,18)(13,20,29,35)(15,25,33,26)(22,28,23,34)(24,30,36,31);(1,2,3,4)(5,7,10,21,12,23,11,22)(6,13,16,30,18,31,17,29)(8,24,25,36,9,20,26,35)(14,32,33,34,27,28,15,19);(5,8,12,9)(6,14,18,27)(7,20,23,24)(10,25,11,26)(13,28,31,32)(15,16,33,17)(19,29,34,30)(21,35,22,36)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=i,d^2=h,f^2=i,g^2=i,b^a=bd,d^a=dg,f^a=fh,g^a=gh,h^a=hi,c^b=ci,e^b=eghi,e^d=ehi,f^e=fi,g^e=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72293126, #72293186, #72293215 |
| maximal normal subgroup of | #72293126, #72293186, #72293215 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59058);