Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60460 |
| 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 | #60450 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2638 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)(5,6,10,16,11,17,26,15)(7,18,20,12,21,28,36,29)(8,13,24,27,23,31,35,32)(9,14,22,34,25,30,19,33);(5,7,10,20,11,21,26,36)(6,12,15,18,17,29,16,28)(8,22,24,25,23,19,35,9)(13,30,32,34,31,14,27,33);(2,4)(5,8)(6,13)(7,19)(9,20)(10,24)(11,23)(12,14)(15,32)(16,27)(17,31)(18,33)(21,22)(25,36)(26,35)(28,34)(29,30);(5,9,10,22,11,25,26,19)(6,14,16,34,17,30,15,33)(7,8,20,24,21,23,36,35)(12,27,28,31,29,32,18,13);(8,23)(9,25)(13,31)(14,30)(19,22)(24,35)(27,32)(33,34)Polycyclic presentation:
2^9|a^2=h,b^2=f,d^2=h,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ci,e^c=ei,e^d=ei116 conjugacy classes — view the full character table.
| maximal subgroup of | #72264764, #72264790, #72264869, … |
| maximal normal subgroup of | #72264764, #72264790, #72264869, … |
| maximal quotient of | #72309747, #72309778, #72310066, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6280169);