Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32700 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2760 |
| commutator | C24 | C23×C4 |
| Frattini, prime radical, cosocle | C25 | C24 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
918 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)(7,15)(8,12,21,27)(9,13,23,29)(10,14)(11,18)(16,24,19,30)(17,28,22,25)(20,33)(26,34)(31,36)(32,35);(5,7)(6,11,14,15)(8,19,21,16)(9,22,23,17)(10,18)(12,24)(13,28)(20,31)(25,29)(26,36,33,35)(27,30)(32,34);(5,8,10,21)(6,12)(7,16,18,19)(9,20,23,34)(11,24)(13,26)(14,27)(15,30)(17,31,22,32)(25,35)(28,36)(29,33);(5,9)(6,13,14,29)(7,17)(8,20)(10,23)(11,25,15,28)(12,26,27,33)(16,31)(18,22)(19,32)(21,34)(24,35,30,36)Polycyclic presentation:
2^9|a^2=h,b^2=ef,c^2=egi,d^2=eg,b^a=be,c^a=cf,d^a=def,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,f^c=fi,h^c=hi,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72285059, #72285164, #72285228, … |
| maximal normal subgroup of | #72285059, #72285164, #72285228, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295208);