Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514148 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3109 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6)(7,12)(8,20)(9,11)(10,15)(13,21)(14,24)(16,27)(17,32)(18,25)(19,28)(22,29)(23,30)(26,31)(33,35)(34,36);(1,2)(3,4)(6,11)(7,16)(9,24)(13,20)(14,15)(17,31)(18,19)(21,22)(25,27)(29,30)(32,35)(33,34);(2,4);(2,4)(5,7,10,19)(6,12,15,28)(8,17,23,33)(9,18,24,16)(11,25,14,27)(13,26,29,36)(20,32,30,35)(21,31,22,34);(5,8)(6,13)(7,17)(9,22)(10,23)(11,20)(12,26)(14,30)(15,29)(16,31)(18,34)(19,33)(21,24)(25,32)(27,35)(28,36)Polycyclic presentation:
2^9|d^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,f^b=fi,h^b=hi,h^d=hi,f^e=fi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7421350);