Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993915 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#124 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2085 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)(9,10)(11,13)(12,14)(15,16)(17,18,21,24)(19,22,23,20);(9,10)(11,15)(12,14)(13,16)(17,19,20,18,21,23,22,24);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15);(11,16)(13,15)(17,19)(18,22)(20,24)(21,23);(1,4,5,3)(2,6,7,8)(10,14)(13,15)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=gh,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,d^c=dh,f^d=fi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72251213, #72256891, #72256924, … |
| maximal normal subgroup of | #72251213, #72256891, #72256924, … |
| maximal quotient of | #72307717 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6997355);