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 | C2×#2213 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #1772 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
630 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,8,3,14,13)(4,16,20,29,21,31,19,27)(5,10,7,24,23,12,26,22)(9,25,17,18,15,30,28,32);(1,3)(2,8)(4,17,21,28)(5,22)(6,11)(7,12)(9,19,15,20)(10,26)(13,14)(16,30,31,25)(18,27,32,29)(23,24);(1,4)(2,9)(3,15)(5,18)(6,19)(7,25)(8,21)(10,27)(11,28)(12,29)(13,17)(14,20)(16,22)(23,32)(24,31)(26,30);(1,5,14,26,8,23,6,7)(2,10,13,22,3,12,11,24)(4,18,20,30,21,32,19,25)(9,27,17,16,15,29,28,31)Polycyclic presentation:
2^9|a^2=g,b^2=egi,d^2=ei,e^2=i,f^2=h,g^2=i,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, #72214250, #72214308, … |
| maximal normal subgroup of | #72214039, #72214250, #72214308, … |
| maximal quotient of | #72216436, #72227062, #72227064, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405538);