Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469956 |
| 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×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1840 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6)(7,10)(8,15)(9,11)(12,16)(13,17)(14,20)(18,19)(21,22,27,24)(23,31,32,36)(25,30,26,29)(28,35,34,33);(5,7)(6,10)(8,14)(9,13)(11,17)(12,18)(15,20)(16,19)(21,23,27,32)(22,28,24,34)(25,33,26,35)(29,31,30,36);(1,2,3,4)(6,11)(7,13)(12,15)(14,19)(21,24,27,22)(23,28,32,34)(25,30,26,29)(31,35,36,33);(7,13)(10,17)(14,19)(18,20)(22,24)(23,32)(29,30)(33,35);(5,8,9,16)(6,12,11,15)(7,14,13,19)(10,18,17,20)(21,25,27,26)(22,29,24,30)(23,33,32,35)(28,31,34,36)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=gh,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6405622);