Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2760365 |
| 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 | 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)(8,10)(11,12)(13,14,18,16)(15,20,17,19)(21,22,25,26)(23,27,31,29)(24,32,33,28)(30,35,36,34);(5,6,9,7)(8,10,12,11)(13,15,18,17)(14,19,16,20)(22,26)(27,29)(28,32)(34,35);(6,7)(10,11)(14,16)(19,20)(21,23,25,31)(22,27,26,29)(24,30,33,36)(28,34,32,35);(1,2)(3,4)(5,7,9,6)(8,10,12,11)(13,16,18,14)(15,20,17,19)(21,22)(23,29)(24,32)(25,26)(27,31)(28,33)(30,34)(35,36);(5,8,9,12)(6,10,7,11)(13,17,18,15)(14,19,16,20)(21,24,25,33)(22,28,26,32)(23,30,31,36)(27,34,29,35)Polycyclic presentation:
2^9|a^2=fh,b^2=g,c^2=fg,d^2=g,e^2=f,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, 2668126);