Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500401 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1914 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6,9,12)(7,10,13,16)(8,11,15,18)(14,17,19,20)(21,22,25,28)(23,29,30,36)(24,33,31,27)(26,34,35,32);(2,4)(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(23,30)(26,35)(29,36)(32,34);(7,13)(8,15)(10,16)(11,18)(21,23,31,34,25,30,24,32)(22,26,33,36,28,35,27,29);(5,8,9,15)(6,11,12,18)(7,14,13,19)(10,17,16,20);(7,13)(10,16)(14,19)(17,20)Polycyclic presentation:
2^9|a^2=h,c^2=gi,d^2=hi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ehi,d^c=dhi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6433808);