Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5955303 |
| automorphism | Aut(G) order 16777216 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2098 conjugacy classes of subgroups; only the 690 normal subgroups are shown.
This group has 2098 conjugacy classes of subgroups — too many to draw. Only its 690 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,3,4)(6,10)(11,12)(13,14,17,16)(15,19,20,18)(23,29)(24,31)(26,32)(27,34);(1,2)(3,4)(13,15,17,20)(14,18,16,19)(23,29)(26,32)(30,35)(33,36);(5,6,9,10)(7,11,8,12)(13,15)(14,18)(16,19)(17,20)(21,22,25,28)(23,26,29,32)(24,27,31,34)(30,33,35,36);(5,7,9,8)(6,11,10,12)(13,14,17,16)(15,18,20,19)(21,23,25,29)(22,26,28,32)(24,30,31,35)(27,33,34,36);(5,8,9,7)(6,12,10,11)(13,16,17,14)(15,19,20,18)(21,24)(22,27)(23,30)(25,31)(26,33)(28,34)(29,35)(32,36)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=ghi,d^2=gi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,d^b=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5863702);