Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7092 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #331 |
| commutator | C4×C8 | C42 |
| Frattini, cosocle | C2×C82 | C22 |
| LCS | G ▷ C4×C8 ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C82 ▷ C42 ▷ C22 ▷ C1 | |
200 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 200 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 512 |
A minimal faithful permutation representation — 2 generators on 36 points:
(1,2,3,4)(6,11)(7,15)(8,17)(9,16)(12,13)(14,20)(18,19)(21,22)(23,26)(24,33)(25,29)(27,30)(28,31)(32,35)(34,36);(5,6,8,12,9,13,7,11,10,14,15,19,16,18,17,20)(22,26)(23,30)(24,31)(27,28)(29,35)(32,36)(33,34)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fh,e^2=fgi,f^2=h,g^2=i,b^a=bc,c^a=cf,e^a=efgh,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=gi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72216562, #72225188, #72225193, … |
| maximal normal subgroup of | #72216562, #72225188, #72225193, … |
| maximal quotient of | #72216560, #72225599, #72227044 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 260);