Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994043 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#123 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2199 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6)(7,12)(8,11)(9,10)(13,14)(15,18)(16,19)(17,20)(21,23)(22,26)(24,27)(25,28);(5,6,7,11,9,10,8,12)(13,14)(15,21)(16,19)(17,20)(18,23)(22,27)(24,26)(25,28);(2,4)(6,10)(11,12)(13,15,17,23)(14,18,20,21)(16,24,25,22)(19,27,28,26);(5,6)(7,12)(8,11)(9,10)(13,16,17,25)(14,19,20,28)(15,22,23,24)(18,26,21,27);(5,7,9,8)(6,11,10,12)(14,20)(18,21)(19,28)(26,27)Polycyclic presentation:
2^9|b^2=fi,c^2=hi,d^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=ehi,d^c=dh,e^d=ei,f^d=fi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72282798, #72282891, #72307669, … |
| maximal normal subgroup of | #72282798, #72282891, #72307669, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7032830);