Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1870 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)(8,9)(10,12)(13,14,18,16)(15,19,17,20)(21,22)(23,29)(24,32)(25,31)(26,27)(28,34)(30,35)(33,36);(5,7,10,8)(6,11,12,9)(13,15,18,17)(14,19,16,20)(22,27)(25,29)(28,32)(33,35);(6,12)(7,8)(13,16,18,14)(15,19,17,20)(21,23)(22,25)(24,30)(26,31)(27,29)(28,35)(32,33)(34,36);(1,2)(3,4)(5,8,10,7)(6,11,12,9)(13,15,18,17)(14,20,16,19)(21,24,26,34)(22,28,27,32)(23,30,31,36)(25,35,29,33);(5,9,10,11)(6,7,12,8)(13,17,18,15)(14,19,16,20)(21,25)(22,23)(24,33)(26,29)(27,31)(28,36)(30,32)(34,35)Polycyclic presentation:
2^9|a^2=fgh,b^2=h,c^2=fgh,d^2=fh,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5544848);