Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #422131 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2416 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2018 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1230 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,15)(8,22)(9,27)(10,17)(11,16)(12,18)(13,20)(14,25)(19,34)(21,29)(23,31)(24,33)(26,32)(28,36)(30,35);(1,2)(3,4)(5,7,10,21,12,20,11,9)(6,13,16,29,18,15,17,27)(8,23,25,35,24,36,26,19)(14,30,22,31,32,34,33,28);(2,4)(5,8)(6,14)(7,19)(9,23)(10,26)(11,25)(12,24)(13,28)(15,31)(16,33)(17,22)(18,32)(20,35)(21,36)(27,30)(29,34);(5,9,10,20,12,21,11,7)(6,15,16,29,18,13,17,27)(8,23,26,35,24,36,25,19)(14,31,33,34,32,28,22,30)Polycyclic presentation:
2^9|b^2=f,d^2=ef,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 336666);