Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994107 |
| 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×#459 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1654 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,4)(3,7,8,6)(11,15)(14,16)(18,22)(19,20)(23,24);(1,3,5,8)(2,6,4,7)(10,13)(11,15);(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,24)(21,23);(1,4,5,2)(3,6,8,7)(10,13)(14,16);(9,11,12,15)(10,14,13,16)(17,19,21,20)(18,22,23,24)Polycyclic presentation:
2^9|a^2=f,b^2=f,d^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ehi,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72251213, #72256852, #72256897, … |
| maximal normal subgroup of | #72251213, #72256852, #72256897, … |
| maximal quotient of | #72308531 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7053756);