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 | |
6819 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)(7,13)(8,12)(9,11)(10,14)(15,19)(16,18)(17,20)(21,22)(23,27)(24,28)(25,29)(26,30)(31,35)(32,34)(33,36);(2,4)(21,23,25,30)(22,26,29,27)(24,31,33,32)(28,34,36,35);(5,7)(6,10)(8,15)(9,14)(11,13)(12,17)(16,20)(18,19)(21,23)(22,27)(24,32)(25,30)(26,29)(28,34)(31,33)(35,36);(7,14)(10,13)(15,20)(17,19)(23,30)(26,27)(31,32)(34,35);(6,11)(10,13)(12,18)(17,19)(21,24,25,33)(22,28,29,36)(23,31,30,32)(26,34,27,35);(5,8,9,16)(6,12,11,18)(7,15,14,20)(10,17,13,19)Polycyclic presentation:
2^9|a^2=gi,b^2=i,e^2=i,f^2=h,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, 7615226);