Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99923 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
829 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6)(7,13,15,10)(8,16)(9,12)(11,17)(14,19,20,18)(21,22,27,32)(23,33,34,36)(24,28,35,29)(25,31,26,30);(1,2)(3,4)(5,7)(6,10)(8,14)(9,15)(11,19)(12,13)(16,18)(17,20)(21,23)(22,28)(24,26)(25,35)(27,34)(29,32)(30,36)(31,33);(2,4)(5,8)(6,11,12,16)(7,14)(9,17)(10,18,13,19)(15,20)(21,24,25,34,27,35,26,23)(22,29,30,36,32,28,31,33)Polycyclic presentation:
2^9|a^2=g,c^2=d,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,d^b=di,e^b=eh,f^b=fi,g^b=ghi,e^c=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 26455);