Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #405621 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14915 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2008 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1333 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)(3,11)(4,15)(5,18)(9,25)(10,21)(12,28)(13,14)(16,17)(19,23)(22,31)(30,32);(1,2,5,7,6,10,18,21)(3,8,11,23,14,24,13,19)(4,9,15,20,17,25,16,26)(12,22,27,31,28,32,29,30);(1,3)(2,8)(4,12)(5,13)(6,14)(7,19)(9,22)(10,24)(11,18)(15,29)(16,27)(17,28)(20,30)(21,23)(25,32)(26,31);(1,4,5,15,6,17,18,16)(2,9,7,20,10,25,21,26)(3,12,13,29,14,28,11,27)(8,22,19,30,24,32,23,31)Polycyclic presentation:
2^9|b^2=h,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,g^b=gi,f^c=fi,h^c=hi,f^d=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72339024, #72339029, #72339032 |
| maximal normal subgroup of | #72339024, #72339029, #72339032 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 312817);