Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23892 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2403 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13967 |
| prime radical | #60387 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2349 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,13,8,14,7,12)(3,15,20,28,19,29,18,9)(4,10,22,24,11,5,23,25)(16,17,27,26,31,30,21,32);(1,3,6,20,8,19,7,18)(2,9,13,15,14,28,12,29)(4,21,22,16,11,27,23,31)(5,17,25,26,10,30,24,32);(1,4)(2,10)(3,16)(5,14)(6,22)(7,23)(8,11)(9,26)(12,25)(13,24)(15,30)(17,29)(18,21)(19,31)(20,27)(28,32);(1,5)(2,11)(3,17)(4,14)(6,25)(7,24)(8,10)(9,27)(12,22)(13,23)(15,31)(16,29)(18,32)(19,30)(20,26)(21,28)Polycyclic presentation:
2^9|a^2=h,b^2=h,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,g^c=gi,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72215968, #72219683, #72219684, … |
| maximal normal subgroup of | #72215968, #72219683, #72219684, … |
| maximal quotient of | C2×#364599 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271795);