Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7008 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 36 points:
(1,2)(3,4)(5,6)(7,10)(8,15)(9,12)(11,16)(13,19)(14,18)(17,20);(2,4)(5,7,9,14)(6,10,12,18)(8,13,16,20)(11,17,15,19)(21,22)(23,29)(24,28)(25,27)(26,30)(31,35)(32,34)(33,36);(5,8)(6,11)(7,13)(9,16)(10,17)(12,15)(14,20)(18,19)(21,23)(22,26)(24,31)(25,30)(27,29)(28,33)(32,36)(34,35);(8,16)(11,15)(13,20)(17,19)(22,27)(23,30)(28,34)(31,36);(6,12)(10,18)(11,15)(17,19)(23,30)(26,29)(31,36)(33,35);(21,24,25,32)(22,28,27,34)(23,31,30,36)(26,33,29,35)Polycyclic presentation:
2^9|b^2=h,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,d^c=dhi,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7615403);