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 | |
4312 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)(7,8)(9,10)(11,12)(13,14,17,19)(15,20,18,16)(21,22,25,26)(23,24,27,28);(6,10)(8,11)(13,15)(14,16)(17,18)(19,20)(22,26)(24,28);(5,7)(6,11)(8,10)(9,12)(13,14)(15,20)(16,18)(17,19)(21,22,25,26)(23,24,27,28);(1,2)(3,4)(5,7)(6,8)(9,12)(10,11)(13,14)(15,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28);(5,8)(6,12)(7,10)(9,11)(13,16)(14,18)(15,19)(17,20)(21,24)(22,27)(23,26)(25,28)Polycyclic presentation:
2^9|a^2=h,c^2=fg,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 | #72232610, #72232618, #72232635, … |
| maximal normal subgroup of | #72232610, #72232618, #72232635, … |
| maximal quotient of | #72328421, #72328754, #72329987, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1326648);