Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #401788 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14823 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1176 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,10,5,11,19,8)(3,7,15,22,14,23,12,21)(4,16,18,24,17,9,25,26)(13,28,30,31,29,20,27,32);(1,3)(2,7)(4,13)(5,14)(6,12)(8,22)(9,20)(10,21)(11,23)(15,19)(16,28)(17,29)(18,27)(24,32)(25,30)(26,31);(2,8)(3,12)(6,19)(9,24)(10,11)(13,27)(14,15)(16,26)(18,25)(21,22)(29,30)(31,32);(1,4,5,17)(2,9)(3,13,14,29)(6,18,19,25)(7,20)(8,24)(10,26)(11,16)(12,27,15,30)(21,31)(22,32)(23,28)Polycyclic presentation:
2^9|a^2=g,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,g^b=gi,f^c=fi,g^c=gi,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72319788 |
| maximal normal subgroup of | #72319788 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 308984);