Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438967 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1725 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,5)(3,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,18,20,23,21,24,22,19);(2,5)(6,8)(17,18)(19,20)(21,24)(22,23);(10,13)(14,16)(17,19,20,18,21,23,22,24);(1,3,4,7)(2,6,5,8)(17,19,20,18,21,23,22,24);(9,11,12,15)(10,14,13,16)(17,20,21,22)(18,19,24,23)Polycyclic presentation:
2^9|a^2=f,c^2=hi,d^2=f,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei,e^d=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72313303, #72313540, #72313729 |
| maximal normal subgroup of | #72313303, #72313540, #72313729 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6369966);