Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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 | |
2122 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)(8,10)(9,12)(13,14,18,16)(15,19,17,20)(21,22)(23,29)(24,26)(25,28)(27,31)(30,35)(32,33)(34,36);(5,7,10,9)(6,11,8,12)(13,15,18,17)(14,19,16,20)(22,24)(25,33)(27,29)(30,36);(5,6,10,8)(7,12,9,11)(13,15,18,17)(14,20,16,19)(21,23,26,31)(22,27,24,29)(25,30,33,36)(28,34,32,35);(1,2)(3,4)(5,8,10,6)(7,11,9,12)(13,16,18,14)(15,19,17,20)(21,24,26,22)(23,27,31,29)(25,28,33,32)(30,35,36,34);(5,9,10,7)(6,11,8,12)(13,17,18,15)(14,19,16,20)(21,25,26,33)(22,28,24,32)(23,30,31,36)(27,34,29,35)Polycyclic presentation:
2^9|a^2=fgh,b^2=fg,c^2=g,d^2=g,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,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, 2978252);