Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #375190 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14152 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60454 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1502 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,12,8,11,6,22)(3,13,15,18,16,9,23,4)(5,19,21,26,20,10,24,27)(14,25,31,32,29,28,30,17);(1,3,6,15,8,16,7,23)(2,9,22,4,11,13,12,18)(5,14,21,30,20,29,24,31)(10,25,27,17,19,28,26,32);(1,4,7,13,8,18,6,9)(2,3,12,15,11,16,22,23)(5,17,24,28,20,32,21,25)(10,14,26,30,19,29,27,31);(1,5)(2,10)(3,14)(4,17)(6,21)(7,24)(8,20)(9,25)(11,19)(12,26)(13,28)(15,30)(16,29)(18,32)(22,27)(23,31)Polycyclic presentation:
2^9|a^2=h,b^2=eg,c^2=fh,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,h^b=hi,e^c=ei,f^d=fi,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72319359, #72319399, #72319453 |
| maximal normal subgroup of | #72319359, #72319399, #72319453 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 282388);