Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806903 |
| 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×#127 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1726 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)(5,6,9,12)(7,13,15,10)(8,16,17,11)(14,18,20,19)(21,22,25,27)(23,31,33,28)(24,29,34,35)(26,30,32,36);(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20)(22,27)(23,24)(26,32)(28,29)(31,35)(33,34);(1,2,3,4)(21,23,32,34,25,33,26,24)(22,28,36,29,27,31,30,35);(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19)(22,27)(23,33)(24,34)(30,36);(7,15)(10,13)(14,20)(18,19)(21,24,26,33,25,34,32,23)(22,29,30,28,27,35,36,31)Polycyclic presentation:
2^9|a^2=f,c^2=gh,d^2=fi,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6722460);