Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | |
1855 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)(5,6,9,10)(7,12,11,8)(13,14)(15,20)(16,19)(17,18)(21,22)(23,29)(24,28)(25,26)(27,30)(31,36)(32,35)(33,34);(6,10)(7,11)(13,15,18,16)(14,19,17,20)(22,26)(23,30)(28,34)(31,32);(6,10)(8,12)(13,16,18,15)(14,19,17,20)(21,23)(22,27)(24,32)(25,30)(26,29)(28,36)(31,33)(34,35);(1,2)(3,4)(5,7,9,11)(6,8,10,12)(21,22)(23,27)(24,28)(25,26)(29,30)(31,35)(32,36)(33,34);(5,8)(6,11)(7,10)(9,12)(13,17,18,14)(15,20,16,19)(21,24,25,33)(22,28,26,34)(23,31,30,32)(27,35,29,36)Polycyclic presentation:
2^9|a^2=gh,b^2=fg,c^2=fg,d^2=gh,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4513490);