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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2004 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)(5,6,8,10)(7,11,12,9)(21,22)(23,27)(24,26)(25,32)(28,33)(29,31)(30,36)(34,35);(7,12)(9,11)(13,14,17,16)(15,19,20,18)(22,24)(23,29)(28,32)(30,35);(5,6)(7,9)(8,10)(11,12)(13,15,17,20)(14,18,16,19)(21,23,26,29)(22,27,24,31)(25,30,33,35)(28,34,32,36);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(21,24,26,22)(23,27,29,31)(25,28,33,32)(30,36,35,34);(5,6)(7,9)(8,10)(11,12)(13,16,17,14)(15,19,20,18)(21,25,26,33)(22,28,24,32)(23,30,29,35)(27,34,31,36)Polycyclic presentation:
2^9|a^2=g,b^2=fh,c^2=h,d^2=f,e^2=h,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, 2672258);