Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #233521 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#127 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#3184 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1672 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(6,10)(7,8)(11,12)(13,14)(15,23)(16,19)(17,22)(18,21)(20,25)(24,28)(26,27);(2,4)(5,6)(7,12)(8,10)(9,11)(14,19)(16,17)(20,27)(21,22)(23,28)(24,25);(6,10)(7,8)(11,12)(13,15,18,26)(14,20,21,28)(16,25,17,24)(19,27,22,23);(6,11)(10,12)(14,21)(19,22)(20,28)(23,27)Polycyclic presentation:
2^9|c^2=i,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,d^b=dhi,e^b=eh,f^b=fi,g^b=ghi,e^c=eh,f^c=fi,g^c=ghi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72224710, #72224723, #72224725, … |
| maximal normal subgroup of | #72224710, #72224723, #72224725, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 140726);