Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6348967 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | #60558 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
3957 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,6)(4,7)(5,8)(9,10)(11,17)(12,20)(13,15)(14,18)(16,22)(19,24)(21,23);(2,6)(3,4)(7,8)(9,11,13,19)(10,14,15,23)(12,21,22,18)(16,24,20,17);(2,7)(3,4)(6,8);(2,8)(6,7);(2,8)(6,7)(10,15)(14,23)(16,20)(17,24)Polycyclic presentation:
2^9|b^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,g^a=gi,f^b=fh,g^b=gi,f^c=fi,g^c=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72238937, #72238966, #72249817, … |
| maximal normal subgroup of | #72238937, #72238966, #72249817, … |
| maximal quotient of | #72257864 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6256170);