Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418871 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2697 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #16014 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
948 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,11)(3,13)(4,14,19,30)(5,17)(6,23,25,12)(8,16)(9,24,28,20)(10,22)(15,27)(18,32)(21,29)(26,31);(1,3)(2,8)(4,17)(5,14)(6,15)(7,16)(9,26)(10,19)(11,13)(12,27)(18,23)(20,31)(21,28)(22,30)(24,29)(25,32);(1,4)(2,9)(3,10)(5,18)(6,24)(7,19)(8,26)(11,28)(12,30)(13,21)(14,23)(15,31)(16,17)(20,25)(22,27)(29,32);(1,5,6,21,7,22,25,26)(2,10,23,31,11,17,12,29)(3,14,15,28,16,30,32,9)(4,18,24,13,19,27,20,8)Polycyclic presentation:
2^9|a^2=gh,d^2=ef,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,h^b=hi,e^c=ei,f^c=fi,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72318864, #72318890, #72318969, … |
| maximal normal subgroup of | #72318864, #72318890, #72318969, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 358984);