Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| 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 | |
2090 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)(7,9)(8,11)(10,12)(13,14,17,19)(15,16,20,18)(21,22)(23,29)(24,32)(25,31)(26,27)(28,35)(30,36)(33,34);(8,12)(10,11)(13,14,17,19)(15,16,20,18)(22,27)(25,29)(28,32)(34,36);(6,7)(10,11)(13,15,17,20)(14,18,19,16)(21,23,26,31)(22,25,27,29)(24,33,35,30)(28,34,32,36);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,15)(14,16)(17,20)(18,19)(21,24,26,35)(22,28,27,32)(23,30,31,33)(25,36,29,34);(5,8)(6,10)(7,11)(9,12)(13,16)(14,15)(17,18)(19,20)(21,25)(22,23)(24,34)(26,29)(27,31)(28,33)(30,32)(35,36)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=fh,d^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4052541);