Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996335 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1780 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)(3,5)(4,6)(7,8)(10,13)(11,15)(17,18,23,24,22,19,20,21);(1,3,4,7)(2,5,6,8)(10,13)(11,15)(17,19,22,18)(20,21,23,24);(2,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,20,22,23)(18,24,19,21);(10,13)(14,16)(17,21,20,19,22,24,23,18);(9,11,12,15)(10,14,13,16)Polycyclic presentation:
2^9|a^2=fi,b^2=gh,c^2=i,d^2=f,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72251174, #72251275, #72256888, … |
| maximal normal subgroup of | #72251174, #72251275, #72256888, … |
| maximal quotient of | #72264540 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7091554);