Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1907 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)(5,6,9,12)(7,10,14,18)(8,11,16,20)(13,17,15,19)(21,22,25,30,26,31,34,27)(23,28,32,33,29,35,36,24);(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(22,27)(24,28)(25,34)(30,31)(32,36)(33,35);(1,2,3,4)(5,8)(6,11)(7,13)(9,16)(10,17)(12,20)(14,15)(18,19)(21,23)(22,28)(24,27)(25,32)(26,29)(30,33)(31,35)(34,36);(8,16)(11,20)(13,15)(17,19)(21,24,25,28,26,33,34,35)(22,29,30,36,31,23,27,32);(6,12)(7,14)(11,20)(13,15)(23,29)(24,33)(28,35)(32,36)Polycyclic presentation:
2^9|a^2=fh,c^2=g,d^2=f,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6949655);