Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23998 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2404 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13956 |
| prime radical | #62815 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2533 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,13,8,12,6,22)(3,9,16,27,14,28,15,30)(4,17,18,10,19,26,23,31)(5,11,21,25,20,32,24,29);(1,3)(2,9,12,28)(4,5)(6,16)(7,15)(8,14)(10,29,31,25)(11,26,32,17)(13,30,22,27)(18,24)(19,20)(21,23);(1,4)(2,10)(3,5)(6,23)(7,18)(8,19)(9,25)(11,30)(12,31)(13,26)(14,20)(15,24)(16,21)(17,22)(27,32)(28,29);(1,5)(2,11,12,32)(3,4)(6,21)(7,24)(8,20)(9,26,28,17)(10,30,31,27)(13,29,22,25)(14,19)(15,18)(16,23)Polycyclic presentation:
2^9|a^2=h,b^2=g,d^2=g,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,f^c=fi,e^d=ei,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72258084, #72261720, #72280350, … |
| maximal normal subgroup of | #72258084, #72261720, #72280350, … |
| maximal quotient of | #72303572, #72303824 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271172);