Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #461255 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16384 |
| 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 | |
841 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)(3,14,19,31)(4,11)(5,10)(6,12)(7,23)(8,13)(9,17,26,18)(15,24,20,30)(16,25,32,29)(21,27)(22,28);(1,3,7,18,8,19,6,17)(2,9,12,14,13,26,23,31)(4,20,5,16,22,15,21,32)(10,24,11,25,27,30,28,29);(1,4)(2,10)(3,15)(5,6)(7,21)(8,22)(9,24)(11,23)(12,28)(13,27)(14,29)(16,18)(17,32)(19,20)(25,31)(26,30);(1,5,7,22,8,21,6,4)(2,11,23,10,13,28,12,27)(3,16,18,15,19,32,17,20)(9,25,31,24,26,29,14,30)Polycyclic presentation:
2^9|a^2=efg,b^2=f,d^2=ei,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,g^b=gi,d^c=de,e^c=ei,f^c=fi,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72251690, #72251695, #72251700, … |
| maximal normal subgroup of | #72251690, #72251695, #72251700, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 368452);