Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417690 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2669 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1557 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)(5,6,11,16,12,18,10,17)(7,19,22,13,23,34,32,35)(8,24,26,29,28,21,27,14)(9,15,30,36,31,20,33,25);(5,7)(6,13)(8,25)(9,29)(10,32)(11,22)(12,23)(14,31)(15,26)(16,34)(17,19)(18,35)(20,27)(21,30)(24,33)(28,36);(2,4)(5,8,12,28)(6,14,18,29)(7,20,23,15)(9,19,31,34)(10,26,11,27)(13,30,35,33)(16,21,17,24)(22,36,32,25);(5,9,10,30,12,31,11,33)(6,15,16,25,18,20,17,36)(7,21,22,29,23,24,32,14)(8,19,26,35,28,34,27,13)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^c=hi,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327121);