Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418518 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2416 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3068 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1647 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)(5,6,11,15)(7,16,18,33)(8,19,24,31)(9,12,26,29)(10,27,25,14)(13,22,32,23)(17,28,35,36)(20,34,21,30);(5,7)(6,12)(8,20)(9,25)(10,26)(11,18)(13,30)(14,33)(15,29)(16,27)(17,23)(19,28)(21,24)(22,35)(31,36)(32,34);(2,4)(5,8)(6,13)(7,17)(9,21)(10,22)(11,24)(12,28)(14,31)(15,32)(16,34)(18,35)(19,27)(20,26)(23,25)(29,36)(30,33);(5,9,25,7,11,26,10,18)(6,12,27,33,15,29,14,16)(8,21,23,17,24,20,22,35)(13,28,19,30,32,36,31,34)Polycyclic presentation:
2^9|a^2=i,d^2=ei,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325717);