Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500701 |
| 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 |
| prime radical | C2×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1882 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,8)(9,11)(10,12)(13,14,18,20,19,21,17,27)(15,16,22,25,23,26,28,24);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15,19,23)(14,16,21,26)(17,28,18,22)(20,25,27,24);(2,4)(5,8,9,10)(6,7,11,12)(13,16)(14,15)(17,25)(18,24)(19,26)(20,28)(21,23)(22,27);(6,11)(7,12)(13,14)(15,16)(17,20)(18,27)(19,21)(22,24)(23,26)(25,28);(13,17,19,18)(14,20,21,27)(15,22,23,28)(16,24,26,25)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,e^b=ei,e^c=ei,g^c=gi,g^d=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7410649);