Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11992 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1555 |
| commutator | C42 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C23 | C2×#100 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
378 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 378 conjugacy classes of subgroups — too many to draw. Only its 108 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 — 3 generators on 36 points:
(1,2,4,3)(5,6,8,12)(7,15,16,27)(9,22,20,35)(10,24,21,14)(11,19,25,17)(13,29,28,23)(18,33,31,26)(30,34,36,32);(5,7,9,18,10,17,23,32,8,16,20,31,21,19,29,34)(6,11,13,26,14,15,22,30,12,25,28,33,24,27,35,36);(1,3,4,2)(7,16)(9,20)(11,25)(13,28)(15,27)(17,19)(22,35)(23,29)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=ei,d^2=g,f^2=gh,h^2=i,b^a=bd,d^a=dg,f^a=fgh,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,f^c=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72224797, #72224810, #72224811, … |
| maximal normal subgroup of | #72224797, #72224810, #72224811, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36113);