Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #420064 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #555 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2420 |
| prime radical | C2×#3067 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1326 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,18)(8,24)(9,16)(10,15)(11,17)(12,21)(13,27)(14,26)(19,29)(20,35)(22,31)(23,32)(25,34)(28,33)(30,36);(5,7)(6,12)(8,25)(9,22)(10,21)(11,23)(13,29)(14,34)(15,32)(16,18)(17,31)(19,28)(20,26)(24,30)(27,36)(33,35);(1,2,3,4)(5,8)(6,13)(7,19,23,25)(9,27)(10,26)(11,28)(12,29,31,34)(14,17)(15,24)(16,33)(18,35,32,30)(20,21,36,22);(5,8)(6,14)(7,20)(9,27)(10,26)(11,28)(12,30)(13,17)(15,33)(16,24)(18,29)(19,21)(22,25)(23,36)(31,35)(32,34)Polycyclic presentation:
2^9|c^2=ef,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330381);