Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #427841 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2778 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15594 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
777 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,10,5,11,6,18)(3,12,19,25,15,8,13,21)(4,9,20,23,16,24,17,26)(14,27,28,31,30,22,29,32);(3,13,15,19)(8,21,12,25)(14,29,30,28)(22,32,27,31);(1,3,5,15)(2,8)(4,14,16,30)(6,19,7,13)(9,22)(10,25)(11,12)(17,28,20,29)(18,21)(23,31)(24,27)(26,32);(1,4,5,16)(2,9)(3,14,15,30)(6,20,7,17)(8,22)(10,26)(11,24)(12,27)(13,28,19,29)(18,23)(21,31)(25,32)Polycyclic presentation:
2^9|a^2=efg,b^2=e,c^2=ef,d^2=ef,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^c=ei,f^c=fi,g^c=gi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72318910, #72318916, #72318981, … |
| maximal normal subgroup of | #72318910, #72318916, #72318981, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 344567);