Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #647656 |
| 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 | C22×#120 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3252 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,8)(9,10)(11,12)(13,14)(15,17)(16,20)(18,19)(22,26)(23,28);(2,4)(6,10)(8,11)(14,19)(17,20);(1,2)(3,4)(13,15,18,16)(14,17,19,20)(21,22,25,26)(23,27,28,24);(5,7)(6,11)(8,10)(9,12)(13,16,18,15)(14,20,19,17)(21,23)(22,27)(24,26)(25,28);(5,8,9,11)(6,7,10,12)(13,17,18,20)(14,15,19,16)(21,24,25,27)(22,23,26,28)Polycyclic presentation:
2^9|c^2=fi,d^2=fgi,e^2=fg,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef104 conjugacy classes — view the full character table.
| maximal subgroup of | #72329152, #72329161, #72329185 |
| maximal normal subgroup of | #72329152, #72329161, #72329185 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 588100);