Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6346762 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
4392 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 24 points:
(5,8)(10,15)(12,13)(16,22)(17,18)(20,21)(23,24);(1,2)(3,5)(4,6)(7,8)(9,10,14,18)(11,16,19,23)(12,15,13,17)(20,22,21,24);(1,3)(2,5)(4,7)(6,8)(9,11)(10,16)(12,20)(13,21)(14,19)(15,22)(17,24)(18,23);(11,19)(16,23)(20,21)(22,24);(9,12,14,13)(10,17,18,15)(11,20,19,21)(16,24,23,22)Polycyclic presentation:
2^9|b^2=h,e^2=h,f^2=h,b^a=bf,c^a=cg,e^a=eh,f^a=fh,e^b=eh,f^b=fh,g^b=gi,d^c=dh,f^c=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72237353, #72307976, C2×#6346767, … |
| maximal normal subgroup of | #72237353, #72307976, C2×#6346767, … |
| maximal quotient of | #72307994, #72308003, #72308007, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6253963);