Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #472803 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16837 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
905 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7)(3,12)(4,16)(5,19)(8,22)(9,25)(14,15)(18,29)(20,24)(21,26)(27,28)(31,32);(1,2,5,10,6,7,19,11)(3,8,14,22,15,20,12,24)(4,9,16,25,18,21,29,26)(13,23,28,31,17,30,27,32);(1,3)(2,8)(4,17)(5,14)(6,15)(7,20)(9,23)(10,22)(11,24)(12,19)(13,18)(16,27)(21,30)(25,31)(26,32)(28,29);(1,4,5,16,6,18,19,29)(2,9,10,25,7,21,11,26)(3,13,12,27,15,17,14,28)(8,23,24,32,20,30,22,31)Polycyclic presentation:
2^9|b^2=egh,d^2=f,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,e^b=ei,h^b=hi,d^c=de,f^c=fi,e^d=ei,g^d=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72338620, #72338627, #72338632 |
| maximal normal subgroup of | #72338620, #72338627, #72338632 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 379999);