Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23990 |
| 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 | |
2541 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,7,12,5,8)(3,9,14,23,15,25,20,21)(4,16,19,26,18,22,17,10)(13,27,28,24,29,30,31,32);(2,8,12,11)(9,23,25,21)(10,22,26,16)(24,30,32,27);(1,3)(2,9)(4,13)(5,20)(6,14)(7,15)(8,21)(10,24)(11,23)(12,25)(16,30)(17,31)(18,29)(19,28)(22,27)(26,32);(1,4)(2,10,12,26)(3,13)(5,17)(6,19)(7,18)(8,22,11,16)(9,24,25,32)(14,28)(15,29)(20,31)(21,27,23,30)Polycyclic presentation:
2^9|a^2=h,b^2=g,d^2=g,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,e^c=ei,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72258084, #72306445, #72306448, … |
| maximal normal subgroup of | #72258084, #72306445, #72306448, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271124);