Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #272667 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1018 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(6,10)(7,11)(8,12)(13,14)(15,23,25,19)(16,20)(17,27,28,21)(18,22)(24,26);(1,2)(3,4)(5,6,9,7)(8,11,12,10)(13,15,18,25)(14,19,22,23)(16,21,26,27)(17,24,28,20);(2,4)(13,16,18,26)(14,20,22,24)(15,21,25,27)(17,23,28,19);(5,7,9,6)(8,10,12,11)(13,17)(14,21)(15,24)(16,23)(18,28)(19,26)(20,25)(22,27)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=hi,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,e^c=ehi,e^d=ei,f^d=fhi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 179893);