Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #456514 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| 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 | |
1455 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)(7,13,8,19)(9,14,25,32)(10,27,20,15)(12,22,29,18)(17,34,21,30)(23,36,35,28)(24,33,26,31);(5,7,10,18,11,8,20,22)(6,12,15,13,16,29,27,19)(9,17,26,23,25,21,24,35)(14,28,33,30,32,36,31,34);(2,4)(5,8)(6,13)(7,11)(9,23)(10,18)(12,15)(14,28)(16,19)(17,26)(20,22)(21,24)(25,35)(27,29)(30,31)(32,36)(33,34);(5,9)(6,14)(7,17)(8,21)(10,26)(11,25)(12,28)(13,30)(15,33)(16,32)(18,23)(19,34)(20,24)(22,35)(27,31)(29,36)Polycyclic presentation:
2^9|a^2=i,b^2=g,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,h^b=hi,d^c=de,e^c=ei,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, 371262);