Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2760365 |
| automorphism | Aut(G) order 1048576 |
| 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 | |
1680 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(7,11)(10,12)(13,14)(15,19)(16,20)(17,18)(21,22,26,27)(23,24,28,31)(25,33,35,29)(30,34,36,32);(2,4)(5,6,9,8)(7,10,11,12)(13,15)(14,16)(17,19)(18,20)(22,27)(24,31)(29,33)(32,34);(1,2)(3,4)(13,16,18,19)(14,15,17,20)(21,23)(22,24)(25,30)(26,28)(27,31)(29,32)(33,34)(35,36);(5,7)(6,10)(8,12)(9,11)(13,17,18,14)(15,19,20,16)(21,24)(22,28)(23,27)(25,34)(26,31)(29,30)(32,35)(33,36);(5,8,9,6)(7,12,11,10)(13,15)(14,19)(16,17)(18,20)(21,25,26,35)(22,29,27,33)(23,30,28,36)(24,32,31,34)Polycyclic presentation:
2^9|a^2=fg,b^2=gi,c^2=g,d^2=g,e^2=fi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2696794);