Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6628896 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C211×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
16908 conjugacy classes of subgroups; only the 946 normal subgroups are shown.
This group has 16908 conjugacy classes of subgroups — too many to draw. Only its 946 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 — 5 generators on 36 points:
(1,2)(3,4)(9,20)(11,22)(13,27)(16,31)(18,21)(23,33)(25,28)(29,32);(5,6)(7,11)(8,12)(9,13)(10,14)(15,23)(16,24)(17,22)(18,25)(19,26)(20,27)(21,28)(29,35)(30,33)(31,34)(32,36);(5,7)(6,11)(8,15)(9,16)(10,17)(12,23)(13,24)(14,22)(18,29)(19,30)(20,31)(21,32)(25,35)(26,33)(27,34)(28,36);(2,4)(5,8,10,19)(6,12,14,26)(7,15,17,30)(9,21,20,18)(11,23,22,33)(13,28,27,25)(16,32,31,29)(24,36,34,35);(5,9,10,20)(6,13,14,27)(7,16,17,31)(8,18,19,21)(11,24,22,34)(12,25,26,28)(15,29,30,32)(23,35,33,36)Polycyclic presentation:
2^9|d^2=i,e^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,g^b=gi,f^c=fi,e^d=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72263037, #72263045, #72263057, … |
| maximal normal subgroup of | #72263037, #72263045, #72263057, … |
| maximal quotient of | #72312486 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536099);