Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3928 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,9)(8,10)(11,12)(13,14,17,16)(15,19,20,18)(21,22,25,28)(23,27,24,26);(6,7)(8,12)(14,16)(18,19)(23,24)(26,27);(6,7)(10,11)(15,20)(18,19)(21,23,25,24)(22,26,28,27);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,15)(14,18)(16,19)(17,20)(21,24,25,23)(22,27,28,26);(5,8,9,12)(6,10,7,11)(13,16,17,14)(15,19,20,18)(21,22)(23,26)(24,27)(25,28)Polycyclic presentation:
2^9|a^2=fg,c^2=h,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
| maximal subgroup of | #72327551, #72327563, #72327590, … |
| maximal normal subgroup of | #72327551, #72327563, #72327590, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4374225);