Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438997 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3262 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(9,10,16,11)(12,15,19,14)(13,17,22,18)(20,21,24,23);(5,6)(7,8)(9,11,15,19,16,10,14,12)(13,18,21,24,22,17,23,20);(2,4)(6,8)(9,12,15,11,16,19,14,10)(13,20,21,18,22,24,23,17);(6,8)(9,13)(10,17)(11,18)(12,20)(14,23)(15,21)(16,22)(19,24);(9,14,16,15)(10,12,11,19)(13,21,22,23)(17,24,18,20)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=f,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dhi,e^b=ei,e^c=ei,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346208);