Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #405284 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14911 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3027 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2239 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,8)(5,6)(9,21)(10,22)(11,20)(12,24)(13,27)(14,15)(17,19)(23,26)(25,30)(31,32);(1,2)(3,12,16,21)(4,10)(5,20)(6,8)(7,11)(9,15,24,14)(13,26,27,31)(17,30)(18,25)(19,22)(23,29,32,28);(1,3)(2,9)(4,13)(5,15)(6,14)(7,16)(8,21)(10,23)(11,24)(12,20)(17,29)(18,27)(19,28)(22,31)(25,32)(26,30);(1,4)(2,10)(3,13)(5,17)(6,19)(7,18)(8,22)(9,23)(11,25)(12,26)(14,28)(15,29)(16,27)(20,30)(21,31)(24,32)Polycyclic presentation:
2^9|b^2=h,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,e^c=ei,g^c=gi,h^c=hi,f^d=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72240437, #72240443, #72240455, … |
| maximal normal subgroup of | #72240437, #72240443, #72240455, … |
| maximal quotient of | #72255625 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 312480);