Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #427883 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15375 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23439 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1223 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,8)(3,12)(4,17)(5,6)(11,21)(13,22)(15,16)(18,25)(19,20)(23,28)(27,30)(29,32);(1,2)(3,13)(4,18)(5,21)(6,8)(7,11)(9,15)(10,19)(12,24)(14,29)(16,22)(17,26)(20,25)(23,31)(27,32)(28,30);(1,3)(2,9)(4,14)(5,12)(6,15)(7,16)(8,22)(10,23)(11,24)(13,21)(17,30)(18,32)(19,27)(20,31)(25,28)(26,29);(1,4)(2,10,11,26)(3,14)(5,19)(6,17)(7,20)(8,18,21,25)(9,23,24,29)(12,27)(13,28,22,32)(15,30)(16,31)Polycyclic presentation:
2^9|d^2=ef,e^2=i,f^2=i,g^2=i,h^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,e^b=ei,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72318430, #72318883, #72318969, … |
| maximal normal subgroup of | #72318430, #72318883, #72318969, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 335079);