Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395409 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#394 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#575 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3500 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,9,14,11,16,10,15)(7,17,19,12,20,27,24,28)(8,13,25,31,23,32,22,30)(18,33,21,26,35,29,34,36);(5,7)(6,12)(8,21)(9,19)(10,24)(11,20)(13,29)(14,27)(15,17)(16,28)(18,22)(23,34)(25,35)(26,30)(31,36)(32,33);(5,8)(6,13)(7,18)(9,22)(10,25)(11,23)(12,26)(14,30)(15,31)(16,32)(17,29)(19,34)(20,35)(21,24)(27,33)(28,36);(2,4)(5,9,11,10)(6,14,16,15)(7,19,20,24)(8,22,23,25)(12,27,28,17)(13,30,32,31)(18,34,35,21)(26,33,36,29)Polycyclic presentation:
2^9|a^2=ef,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,e^c=ei,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72238736, #72264928, #72264941, … |
| maximal normal subgroup of | #72238736, #72264928, #72264941, … |
| maximal quotient of | #72238756 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302607);