Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #422933 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15310 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1935 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7)(4,16)(5,19)(8,24)(10,23)(11,21)(12,28)(13,14)(17,22)(18,20)(25,27)(30,31);(1,2,5,7,6,10,19,23)(3,11,13,21,15,8,14,24)(4,17,16,26,18,22,20,9)(12,29,30,25,31,32,28,27);(1,3)(2,8,10,11)(4,12)(5,14)(6,15)(7,21,23,24)(9,25,26,27)(13,19)(16,28)(17,29,22,32)(18,31)(20,30);(1,4)(2,9)(3,12)(5,20)(6,18)(7,22)(8,25)(10,26)(11,27)(13,28)(14,30)(15,31)(16,19)(17,23)(21,32)(24,29)Polycyclic presentation:
2^9|b^2=e,c^2=f,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,e^c=ei,g^c=gi,e^d=ei,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72213913, #72219485, #72219491, … |
| maximal normal subgroup of | #72213913, #72219485, #72219491, … |
| maximal quotient of | C2×#422933, #72255605, #72255608, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330129);