Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1418353 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #14715 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4184 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)(6,10)(7,12)(13,14)(15,19)(16,17)(18,20)(21,22,26,28)(23,27,25,24);(5,6,9,10)(7,11,12,8)(14,16)(18,19)(21,23)(22,24)(25,26)(27,28);(13,15)(14,18)(16,19)(17,20)(21,24)(22,25)(23,28)(26,27);(1,2)(3,4)(5,7)(6,11)(8,10)(9,12)(13,15)(14,19)(16,18)(17,20)(21,24)(22,23)(25,28)(26,27);(5,8,9,11)(6,12,10,7)(13,16,17,14)(15,19,20,18)(21,25,26,23)(22,27,28,24)Polycyclic presentation:
2^9|a^2=h,b^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,e^d=ef92 conjugacy classes — view the full character table.
| maximal subgroup of | #72293912, C2×#1419405, #72304082, … |
| maximal normal subgroup of | #72293912, C2×#1419405, #72304082, … |
| maximal quotient of | #72293987, C2×#1419405 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1326601);