Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #462393 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16429 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ 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:
(1,2,5,11,7,12,6,8)(3,13,17,25,14,9,16,23)(4,18,22,28,21,24,20,10)(15,27,19,29,30,32,31,26);(2,8)(3,14)(5,6)(9,25)(10,24)(11,12)(13,23)(15,30)(18,28)(20,22)(26,32)(27,29);(1,3,5,16,7,14,6,17)(2,9,11,25,12,13,8,23)(4,19,20,30,21,31,22,15)(10,27,24,29,28,32,18,26);(1,4)(2,10,12,28)(3,15)(5,22)(6,20)(7,21)(8,24,11,18)(9,26,13,29)(14,30)(16,31)(17,19)(23,27,25,32)Polycyclic presentation:
2^9|a^2=eg,c^2=e,d^2=f,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,e^b=ei,h^b=hi,d^c=de,g^c=gi,e^d=ei,g^d=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72251784, #72251792, #72251802, … |
| maximal normal subgroup of | #72251784, #72251792, #72251802, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 369589);