Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #498338 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #23001 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | #33594 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #1772 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
610 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,8,14)(3,15)(4,19)(5,12,22,31)(6,13,9,26)(7,24,23,32)(10,20)(11,18)(16,30)(17,28)(21,29)(25,27);(1,3,9,17,8,4,6,20)(2,10,26,15,14,28,13,19)(5,21,23,18,22,25,7,16)(11,24,27,12,30,32,29,31);(1,4)(2,11)(3,8)(5,18)(6,20)(7,25)(9,17)(10,12)(13,27)(14,30)(15,24)(16,22)(19,32)(21,23)(26,29)(28,31);(1,5,6,7,8,22,9,23)(2,12,13,24,14,31,26,32)(3,16,17,21,4,18,20,25)(10,27,15,30,28,29,19,11)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=e,e^2=i,f^2=h,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ch,d^b=de,e^b=ei,f^b=fi,g^b=gi,f^c=fhi,g^c=gi,h^c=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72214039, #72214217, #72214308, … |
| maximal normal subgroup of | #72214039, #72214217, #72214308, … |
| maximal quotient of | #72216436, #72227061, #72227063, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405534);