Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61058 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #60366 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#408 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6,9,16,10,14,23,15)(7,13,19,25,20,30,18,11)(8,12,22,28,21,29,34,27)(17,26,33,35,31,36,32,24);(5,7,9,19,10,20,23,18)(6,11,16,13,14,25,15,30)(8,17,22,33,21,31,34,32)(12,24,28,26,29,35,27,36);(2,4)(5,8)(6,12)(7,17)(9,22)(10,21)(11,24)(13,26)(14,29)(15,27)(16,28)(18,32)(19,33)(20,31)(23,34)(25,35)(30,36);(5,7)(6,13)(8,17)(9,19)(10,20)(11,15)(12,26)(14,30)(16,25)(18,23)(21,31)(22,33)(24,27)(28,35)(29,36)(32,34);(6,14)(8,21)(11,25)(13,30)(15,16)(17,31)(22,34)(32,33)Polycyclic presentation:
2^9|a^2=h,b^2=h,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,d^b=df,f^b=fi,e^c=ei,f^d=fi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72264869, #72308688, #72308721, … |
| maximal normal subgroup of | #72264869, #72308688, #72308721, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6288655);