Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417601 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2330 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,17)(8,22)(9,15)(10,14)(11,16)(12,20)(13,23)(18,21)(19,27)(24,29)(25,28)(26,30)(31,34)(32,33)(35,36);(1,2)(3,4)(5,6)(7,18)(8,13)(9,14)(10,15)(11,16)(12,20)(17,21)(19,32)(22,23)(24,30)(25,28)(26,29)(27,33)(31,35)(34,36);(5,7)(6,12)(8,19)(9,25)(10,20)(11,21)(13,27)(14,18)(15,17)(16,28)(22,32)(23,34)(24,35)(26,33)(29,31)(30,36);(2,4)(5,8,11,24)(6,13,16,30)(7,19,21,35)(9,23,10,26)(12,27,28,36)(14,22,15,29)(17,31,18,32)(20,33,25,34)Polycyclic presentation:
2^9|d^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,g^b=gi,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, 324798);