Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418884 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15267 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2329 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)(3,12)(4,17,20,27)(5,22)(6,10)(7,11)(8,15)(9,21,26,19)(13,28)(14,23,30,32)(16,24)(18,29,31,25);(1,3,5,15,7,13,6,16)(2,8,10,12,11,24,22,28)(4,18,19,30,20,31,21,14)(9,25,17,32,26,29,27,23);(3,13)(5,6)(10,22)(12,28)(14,30)(17,27)(19,21)(25,29);(1,4)(2,9)(3,14)(5,21)(6,19)(7,20)(8,23)(10,27)(11,26)(12,29)(13,30)(15,31)(16,18)(17,22)(24,32)(25,28)Polycyclic presentation:
2^9|a^2=f,b^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,g^b=gi,e^c=ei,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72213423, #72213776, #72213781, … |
| maximal normal subgroup of | #72213423, #72213776, #72213781, … |
| maximal quotient of | C2×#418884 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326080);