Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| 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, cosocle | C24 | C25 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2155 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)(7,11)(8,12)(9,10)(13,14,18,16)(15,20,17,19)(21,22,25,28)(23,26,29,33)(24,31,32,27)(30,36,35,34);(5,7)(6,8)(9,12)(10,11)(13,14,18,16)(15,19,17,20)(23,29)(26,33)(30,35)(34,36);(6,10)(8,11)(13,15,18,17)(14,19,16,20)(21,23,25,29)(22,26,28,33)(24,30,32,35)(27,34,31,36);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,16,18,14)(15,19,17,20)(21,22)(23,26)(24,27)(25,28)(29,33)(30,34)(31,32)(35,36);(5,8)(6,7)(9,11)(10,12)(13,17,18,15)(14,19,16,20)(21,24,25,32)(22,27,28,31)(23,30,29,35)(26,34,33,36)Polycyclic presentation:
2^9|a^2=fgh,b^2=fg,c^2=fgh,d^2=fg,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5389274);