Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24420 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2410 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13995 |
| prime radical | #60387 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2224 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)(3,8,13,25)(4,15,18,5)(6,9)(7,21)(10,20)(11,29)(12,16)(14,27,22,26)(17,19,23,28)(24,30)(31,32);(1,3,7,22,6,13,20,14)(2,8,21,26,9,25,10,27)(4,16,23,30,18,32,17,11)(5,12,19,24,15,31,28,29);(1,4)(2,5)(3,11)(6,18)(7,23)(8,24)(9,15)(10,28)(12,27)(13,30)(14,32)(16,22)(17,20)(19,21)(25,29)(26,31);(1,5)(2,4)(3,12)(6,15)(7,19)(8,16)(9,18)(10,17)(11,27)(13,31)(14,29)(20,28)(21,23)(22,24)(25,32)(26,30)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,g^b=gi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72213423, #72213889, #72213913, … |
| maximal normal subgroup of | #72213423, #72213889, #72213913, … |
| maximal quotient of | C2×#366636 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 273832);