Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6347305 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3496 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
4148 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,7)(3,8,4,6)(9,10,13,16)(11,17,18,24)(12,20,22,15)(14,21,23,19);(2,6)(3,4)(7,8)(9,11)(10,14)(12,21)(13,18)(15,24)(16,23)(17,20)(19,22);(12,22)(15,20)(17,24)(19,21);(2,7)(6,8);(1,3,5,4)(2,8,7,6)(11,18)(14,23)(17,24)(19,21)Polycyclic presentation:
2^9|a^2=h,e^2=hi,f^2=h,b^a=bf,c^a=cg,d^a=dhi,f^a=fh,e^b=eh,f^b=fh,g^b=gi,f^c=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72257713, #72282423, #72282438, … |
| maximal normal subgroup of | #72257713, #72282423, #72282438, … |
| maximal quotient of | #72264649 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6254533);