Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470067 |
| 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 | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4380 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)(3,5)(4,6)(7,8)(10,16)(11,19)(13,14)(17,18)(21,24)(22,23);(1,3)(2,5)(4,7)(6,8)(9,10)(11,20)(12,21)(13,17)(14,16)(15,18)(19,23)(22,24);(3,7)(5,8)(9,11)(10,12)(13,19)(14,24)(15,21)(16,23)(17,22)(18,20);(3,7)(5,8)(9,12)(10,11)(13,23)(14,22)(15,20)(16,19)(17,24)(18,21);(2,6)(5,8)(9,13,15,14)(10,17,18,16)(11,19,21,24)(12,22,20,23)Polycyclic presentation:
2^9|e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=fi,g^c=gi,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72311985, #72312146, #72312623, … |
| maximal normal subgroup of | #72311985, #72312146, #72312623, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6377267);