Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3180 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)(5,6,8,10)(7,11,12,9)(13,14,16,17)(15,19,20,18)(21,22)(23,25)(24,27)(26,28);(5,7,8,12)(6,9,10,11)(13,14)(15,19)(16,17)(18,20)(21,23,26,27)(22,24,28,25);(6,10)(9,11)(14,17)(18,19)(21,24,26,25)(22,27,28,23);(5,7)(6,9)(8,12)(10,11)(13,15,16,20)(14,18,17,19)(21,25,26,24)(22,27,28,23);(5,6)(7,9)(8,10)(11,12)(13,14)(15,18)(16,17)(19,20)(21,22)(23,24)(25,27)(26,28)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=fg,d^2=h,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dg,e^c=ef104 conjugacy classes — view the full character table.
| maximal subgroup of | #72328698, #72328731, #72328817, … |
| maximal normal subgroup of | #72328698, #72328731, #72328817, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 484125);