Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #456514 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23402 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1193 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)(7,18)(8,14,25,32)(9,16)(10,15)(11,17)(12,20)(13,21)(19,34,35,30)(22,28)(23,27,36,29)(24,33,26,31);(1,2)(3,4)(5,7,10,21,11,22,9,20)(6,12,16,28,17,13,15,18)(8,19,24,23,25,35,26,36)(14,27,31,34,32,29,33,30);(2,4)(5,7)(6,13)(8,23)(9,21)(10,20)(11,22)(12,17)(14,30)(15,18)(16,28)(19,24)(25,36)(26,35)(27,33)(29,31)(32,34);(5,8)(6,14)(7,19)(9,26)(10,24)(11,25)(12,27)(13,29)(15,33)(16,31)(17,32)(18,30)(20,36)(21,23)(22,35)(28,34)Polycyclic presentation:
2^9|a^2=ef,b^2=f,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,h^a=hi,c^b=cg,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 363710);