Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394481 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2472 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14717 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1940 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)(3,13,16,22)(4,17)(5,10)(6,11)(7,12)(8,14,24,20)(9,18)(15,29,30,23)(19,25)(21,26)(27,28,32,31);(3,14,16,20)(8,22,24,13)(15,28,30,31)(23,32,29,27);(1,3)(2,8)(4,15)(5,20)(6,14)(7,16)(9,23)(10,22)(11,13)(12,24)(17,27)(18,31)(19,30)(21,28)(25,32)(26,29);(1,4)(2,9)(3,15)(5,18)(6,21)(7,19)(8,23)(10,25)(11,17)(12,26)(13,27)(14,28)(16,30)(20,31)(22,32)(24,29)Polycyclic presentation:
2^9|a^2=e,b^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,e^c=ei,f^c=fi,g^c=gi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72263012, #72267112, #72280281, … |
| maximal normal subgroup of | #72263012, #72267112, #72280281, … |
| maximal quotient of | #72262963 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301677);