Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10390 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #393 |
| commutator | C25 | C22×C4 |
| Frattini, prime radical, cosocle | C26 | C23 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C26 ◁ G | |
| LECS | G ▷ C26 ▷ C22 ▷ C1 | |
1310 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 1310 conjugacy classes of subgroups — too many to draw. Only its 138 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:
(2,4)(5,6)(7,10)(8,15,17,11)(9,12)(13,20,16,18)(14,19)(21,22)(23,26)(24,27)(25,28)(29,34)(30,33)(31,35)(32,36);(1,2,3,4)(5,7)(6,10)(8,16,17,13)(9,14)(11,18,15,20)(12,19)(21,23)(22,26)(24,29)(25,30)(27,34)(28,33)(31,36)(32,35);(5,8)(6,11)(7,13)(9,17)(10,18)(12,15)(14,16)(19,20)(21,24,25,31)(22,27,28,35)(23,29,30,36)(26,32,33,34)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fhi,c^b=cf,e^b=ei,f^b=fh,d^c=dh,g^c=gh56 conjugacy classes — view the full character table.
| maximal subgroup of | #72235753, #72266477 |
| maximal normal subgroup of | #72235753, #72266477 |
| maximal quotient of | #72235757, #72235763, #72235766, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 23553);