Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
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 | |
4664 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,3,4)(9,10,14,18,16,20,15,19)(11,21,24,13,12,23,22,17);(2,4)(5,6)(7,8)(9,11)(10,13)(12,16)(14,22)(15,24)(17,20)(18,21)(19,23);(6,8)(9,12,16,11)(10,17,20,13)(14,24,15,22)(18,23,19,21);(6,8)(9,13)(10,12)(11,20)(14,23)(15,21)(16,17)(18,22)(19,24);(9,14,16,15)(10,18,20,19)(11,22,12,24)(13,21,17,23)Polycyclic presentation:
2^9|a^2=fi,c^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345232);