Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33605 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×C4×D8 |
| prime radical | #33716 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
760 conjugacy classes of subgroups; only the 234 normal subgroups are shown.
This group has 760 conjugacy classes of subgroups — too many to draw. Only its 234 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) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,10,15,11,17,26,16)(7,18,23,33,20,31,36,34)(8,13,9,25,19,14,27,32)(12,21,29,35,28,24,30,22);(5,7,19,21,10,23,27,35,11,20,8,24,26,36,9,22)(6,12,25,34,16,30,13,31,17,28,32,33,15,29,14,18);(1,2,3,4)(7,20)(12,28)(18,31)(21,24)(22,35)(23,36)(29,30)(33,34);(5,8,10,9,11,19,26,27)(6,13,15,25,17,14,16,32)(7,21,23,35,20,24,36,22)(12,18,29,33,28,31,30,34)Polycyclic presentation:
2^9|a^2=g,b^2=eh,c^2=f,d^2=g,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,c^b=ci,d^b=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72210518 |
| maximal normal subgroup of | #72210518 |
| maximal quotient of | #72210412, #72210416, #72210421 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403310);