Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470395 |
| 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 | C2×#3204 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1896 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)(7,10)(8,11)(9,12)(13,17)(14,18)(15,19)(16,20)(22,28)(25,26)(29,35)(30,36)(31,32)(33,34);(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,22,26,28,27,32,25,31)(23,30,33,36,24,29,34,35);(2,4)(5,8,9,16)(6,11,12,20)(7,13,14,15)(10,17,18,19)(21,23,27,24)(22,29,32,30)(25,34,26,33)(28,35,31,36);(7,14)(10,18)(13,15)(17,19)(21,24,27,23)(22,30,32,29)(25,33,26,34)(28,36,31,35);(6,12)(10,18)(11,20)(17,19)(21,25,27,26)(22,31,32,28)(23,33,24,34)(29,35,30,36)Polycyclic presentation:
2^9|b^2=f,c^2=h,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,e^c=ei,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6384937);