Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2121 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C2×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C23 | C43 |
| Frattini, prime radical, cosocle | C22×#41 | C23 |
| socle | C22 | C2×#342 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
864 conjugacy classes of subgroups; only the 242 normal subgroups are shown.
This group has 864 conjugacy classes of subgroups — too many to draw. Only its 242 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:
(5,6)(7,15)(8,19,20,12)(9,13)(10,14)(11,17,23,29)(16,27)(18,28)(21,32,33,25)(22,26)(24,31,34,36)(30,35);(5,7,9,18,10,16,22,30)(6,11,13,24,14,23,26,34)(8,17,21,31,20,29,33,36)(12,15,25,28,19,27,32,35);(1,2,3,4)(7,16)(11,23)(15,27)(17,29)(18,30)(24,34)(28,35)(31,36)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=g,f^2=i,b^a=bd,d^a=dh,h^a=hi,c^b=ci,e^b=eh,e^d=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72311867, #72311880, #72311884, … |
| maximal normal subgroup of | #72311867, #72311880, #72311884, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53545);