Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6439029 |
| 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 | |
2695 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,4,3)(5,6)(7,12)(8,20)(9,13)(10,15)(11,16)(14,21)(17,27)(18,28)(19,32)(22,35)(23,29)(24,33)(25,30)(26,31)(34,36);(2,3)(5,7)(6,12)(8,21)(9,23)(10,19)(11,18)(13,29)(14,20)(15,32)(16,28)(17,25)(22,26)(24,34)(27,30)(31,35)(33,36);(1,2,4,3)(5,8,10,7,11,19,21,18)(6,12,14,20,16,28,15,32)(9,22,24,17,26,23,25,34)(13,27,30,35,31,36,33,29);(1,3,4,2)(5,9)(6,13)(7,17)(8,22)(10,24)(11,26)(12,27)(14,30)(15,33)(16,31)(18,34)(19,23)(20,35)(21,25)(28,36)(29,32);(7,18)(8,19)(12,28)(17,34)(20,32)(22,23)(27,36)(29,35)Polycyclic presentation:
2^9|a^2=f,c^2=fg,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351505);