Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #420517 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2555 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15476 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1055 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,8,12)(3,13,16,5)(4,17)(6,23,7,11)(9,24,22,15)(10,19)(14,25)(18,28)(20,26)(21,30)(27,32)(29,31);(1,3,7,15,8,16,6,24)(2,9,11,5,12,22,23,13)(4,14,20,28,21,29,19,32)(10,25,17,18,26,31,30,27);(1,4,6,19,8,21,7,20)(2,10,11,17,12,26,23,30)(3,14,15,28,16,29,24,32)(5,18,9,25,13,27,22,31);(1,5,6,9,8,13,7,22)(2,3,11,15,12,16,23,24)(4,18,19,25,21,27,20,31)(10,14,17,28,26,29,30,32)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=e,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^b=ei,h^b=hi,g^c=gi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72318217, #72318940, #72318963, … |
| maximal normal subgroup of | #72318217, #72318940, #72318963, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 340496);