Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #401802 |
| 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 | |
1191 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,11,7,9,6,21)(3,12,15,24,16,8,13,22)(4,17,19,25,18,32,20,10)(14,28,30,26,29,31,27,23);(1,3)(2,8)(4,14)(5,13)(6,15)(7,16)(9,12)(10,23)(11,24)(17,31)(18,29)(19,27)(20,30)(21,22)(25,26)(28,32);(2,9)(3,13)(5,6)(8,22)(10,25)(12,24)(14,27)(15,16)(19,20)(23,28)(26,31)(29,30);(1,4)(2,10,9,25)(3,14)(5,19)(6,20)(7,18)(8,23,12,26)(11,17,21,32)(13,27)(15,30)(16,29)(22,28,24,31)Polycyclic presentation:
2^9|a^2=g,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,g^b=gi,g^c=gi,h^c=hi,f^d=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72216029, #72216130, #72216192, … |
| maximal normal subgroup of | #72216029, #72216130, #72216192, … |
| maximal quotient of | #72237176, C2×#401802, #72255715, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 308998);