Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514160 |
| 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 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2516 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,3)(5,6)(7,15)(8,20)(9,23)(10,14)(11,18)(12,21)(13,24)(16,28)(17,27)(19,26)(22,30)(25,29)(31,32)(33,35)(34,36);(1,2,4,3)(7,16)(9,24)(11,26)(12,29)(13,23)(15,28)(17,33)(18,19)(21,25)(27,35)(31,36)(32,34);(5,7)(6,11)(8,17)(9,16)(10,19)(12,27)(13,26)(14,28)(15,23)(18,24)(20,31)(21,33)(22,34)(25,32)(29,36)(30,35);(5,8,10,22)(6,12,14,29)(7,17,19,34)(9,25,24,21)(11,27,28,36)(13,30,23,20)(15,31,26,35)(16,32,18,33);(1,3,4,2)Polycyclic presentation:
2^9|b^2=f,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,g^b=gi,g^c=gi,h^c=hi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7425989);