Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #429607 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2595 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15404 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1000 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,7,5,8,22)(3,12,14,21,16,25,15,9)(4,17,19,18,20,27,23,10)(13,26,29,28,31,24,30,32);(1,3,6,15,7,16,8,14)(2,9,11,25,5,21,22,12)(4,13,19,30,20,31,23,29)(10,26,17,32,18,24,27,28);(1,4)(2,10)(3,13)(5,18)(6,23)(7,20)(8,19)(9,24)(11,27)(12,28)(14,30)(15,29)(16,31)(17,22)(21,26)(25,32);(1,5,7,2)(3,12)(4,18,20,10)(6,22,8,11)(9,15)(13,28)(14,21)(16,25)(17,19,27,23)(24,29)(26,30)(31,32)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=ef,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,e^c=ei,f^c=fi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72319753, #72319853 |
| maximal normal subgroup of | #72319753, #72319853 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 336803);