Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| automorphism | Aut(G) order 2097152 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1840 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6)(7,12)(8,10)(9,11)(13,14,17,15)(16,18,20,19)(21,22,26,27)(23,28,33,35)(24,29,30,25)(31,32,34,36);(6,8)(9,12)(13,15,17,14)(16,18,20,19)(22,27)(25,29)(28,35)(32,36);(5,7,10,11)(6,9,8,12)(13,16,17,20)(14,18,15,19)(21,23)(22,28)(24,34)(25,32)(26,33)(27,35)(29,36)(30,31);(1,2)(3,4)(5,8,10,6)(7,9,11,12)(21,24)(22,29)(23,31)(25,27)(26,30)(28,32)(33,34)(35,36);(5,9)(6,11)(7,8)(10,12)(21,25,26,29)(22,30,27,24)(23,32,33,36)(28,31,35,34)Polycyclic presentation:
2^9|a^2=fgh,b^2=h,c^2=gh,d^2=g,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,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, 2980406);