Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6439029 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#2002 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2196 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,8)(4,6)(5,7)(9,10,16,11)(12,14,18,15)(13,17,22,19)(20,21,24,23);(1,3,5,4)(2,6,7,8)(9,11,16,10)(12,14,18,15)(13,20,22,24)(17,21,19,23);(9,12,14,11,16,18,15,10)(13,20,21,19,22,24,23,17);(1,4,5,3)(2,6,7,8)(9,13)(10,17)(11,19)(12,20)(14,21)(15,23)(16,22)(18,24);(9,14,16,15)(10,18,11,12)(13,21,22,23)(17,24,19,20)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=g,d^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72213480, #72213590, #72213751 |
| maximal normal subgroup of | #72213480, #72213590, #72213751 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351172);