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 | C2×#2543 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2999 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,12)(8,9)(10,11)(14,17)(15,19)(21,22,25,24)(23,27,28,26);(6,11)(9,12)(22,24)(26,27);(5,7,10,8)(6,12,11,9)(13,14)(15,18)(16,17)(19,20)(21,22,25,24)(23,26,28,27);(1,2)(3,4)(5,8,10,7)(6,12,11,9)(13,15,16,19)(14,18,17,20)(21,23)(22,26)(24,27)(25,28);(5,9,10,12)(6,8,11,7)(13,14)(15,18)(16,17)(19,20)(21,24,25,22)(23,26,28,27)Polycyclic presentation:
2^9|a^2=h,c^2=f,d^2=fgh,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 | #72256708, #72256758, #72256759, … |
| maximal normal subgroup of | #72256708, #72256758, #72256759, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1330118);