Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419800 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2008 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1606 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,9,14,10,15,23,11)(7,16,17,27,20,26,19,12)(8,13,21,28,22,29,34,24)(18,30,31,36,33,35,32,25);(6,11)(7,17)(8,21)(9,23)(14,15)(16,26)(19,20)(22,34)(24,28)(25,35)(30,36)(31,32);(5,7)(6,12)(8,18)(9,19)(10,20)(11,16)(13,25)(14,26)(15,27)(17,23)(21,32)(22,33)(24,30)(28,35)(29,36)(31,34);(2,4)(5,8,9,21,10,22,23,34)(6,13,14,28,15,29,11,24)(7,18,19,32,20,33,17,31)(12,25,26,35,27,36,16,30)Polycyclic presentation:
2^9|a^2=f,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326996);