Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6371542 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C211 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×#386 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | #62852 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
6131 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)(3,11)(4,15,18,8)(5,9)(6,14)(7,10)(12,17,23,21)(13,25)(16,28,29,22)(19,24)(20,27)(26,30,32,31);(1,3,11,2,7,14,6,10)(4,12,17,8,18,23,21,15)(5,13,25,9,19,27,20,24)(16,26,30,22,29,32,31,28);(1,4)(2,8)(3,12)(5,16)(6,21)(7,18)(9,22)(10,15)(11,17)(13,26)(14,23)(19,29)(20,31)(24,28)(25,30)(27,32);(1,5)(2,9)(3,13)(4,16)(6,20)(7,19)(8,22)(10,24)(11,25)(12,26)(14,27)(15,28)(17,30)(18,29)(21,31)(23,32);(5,19)(9,24)(13,27)(16,29)(20,25)(22,28)(26,32)(30,31)Polycyclic presentation:
2^9|a^2=h,b^2=fhi,f^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,g^b=gi,f^c=fi,h^c=hi,e^d=ei113 conjugacy classes — view the full character table.
| maximal subgroup of | #72233774, #72233779, #72240442, … |
| maximal normal subgroup of | #72233774, #72233779, #72240442, … |
| maximal quotient of | C2×#6371543, D4×#1910, #72252478, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6278739);