Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573549 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#447 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1853 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)(3,8)(4,6)(5,7)(17,18)(19,22)(20,23)(21,24);(1,3,5,4)(2,6,7,8)(10,14)(12,15)(18,22)(19,20)(23,24);(9,10)(11,12)(13,14)(15,16)(18,23)(19,20)(22,24);(1,4,5,3)(2,6,7,8)(9,11,13,16)(10,15,14,12);(9,12,13,15)(10,11,14,16)(18,24)(22,23)Polycyclic presentation:
2^9|b^2=fg,d^2=fgh,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,g^a=gi,c^b=ch,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=eh83 conjugacy classes — view the full character table.
| maximal subgroup of | #72282016, #72305298, #72305303 |
| maximal normal subgroup of | #72282016, #72305298, #72305303 |
| maximal quotient of | #72305719, #72305778, #72316324 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6531580);