Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417538 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1826 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,17,19,14)(12,20)(13,16)(15,21)(18,22,24,23);(1,3)(2,5)(4,7)(6,8)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24);(3,7)(5,8)(9,12)(10,15,16,20)(11,18)(13,21)(14,22,17,23)(19,24);(14,17)(22,23)Polycyclic presentation:
2^9|a^2=gh,c^2=e,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,h^b=hi,f^c=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72225693, #72225696, #72225705, … |
| maximal normal subgroup of | #72225693, #72225696, #72225705, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 358119);