Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453999 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2241 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,16)(8,13)(9,12)(10,15)(11,18)(14,23)(17,32)(19,29)(20,30)(21,28)(22,26)(24,27)(25,33)(31,34)(35,36);(5,7)(6,11)(8,17)(9,22)(10,19)(12,16)(13,25)(14,29)(15,26)(18,23)(20,35)(21,34)(24,33)(27,31)(28,36)(30,32);(2,4)(6,12)(8,20)(9,23)(11,16)(13,28)(14,15)(17,33)(18,22)(21,24)(26,29)(31,32)(34,35);(5,8)(6,13)(7,17)(9,24)(10,21)(11,25)(12,27)(14,30)(15,28)(16,31)(18,35)(19,34)(20,23)(22,33)(26,36)(29,32)Polycyclic presentation:
2^9|e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,e^c=ei,f^c=fi,e^d=ei,f^d=fi,g^d=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361195);