Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61106 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×D42 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C2 | C2×#2361 |
| prime radical | #60469 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2453 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,8,11)(3,13)(4,18,20,5)(6,12,7,24)(9,23)(10,25,22,19)(14,15)(16,28)(17,29)(21,31)(26,32)(27,30);(1,3,7,16,8,17,6,23)(2,9,24,13,11,28,12,29)(4,14,25,31,20,32,19,27)(5,21,10,26,18,30,22,15);(1,4)(2,10)(3,14)(5,12)(6,19)(7,25)(8,20)(9,26)(11,22)(13,30)(15,28)(16,31)(17,32)(18,24)(21,29)(23,27);(1,5,6,22,8,18,7,10)(2,4,12,19,11,20,24,25)(3,15,16,21,17,26,23,30)(9,27,13,14,28,31,29,32);(2,11)(5,18)(9,28)(10,22)(12,24)(13,29)(15,26)(21,30)Polycyclic presentation:
2^9|a^2=h,b^2=fh,d^2=f,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,d^b=df,f^b=fi,h^b=hi,g^c=gi,e^d=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72285627 |
| maximal normal subgroup of | #72285627 |
| maximal quotient of | #72286469, #72342647 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6289193);