Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417622 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2408 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2323 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,17)(8,21)(9,15)(10,14)(11,16)(12,19)(13,24)(18,27)(20,28)(22,33)(23,32)(25,31)(26,29)(30,34)(35,36);(5,7)(6,12)(8,22)(9,26)(10,19)(11,20)(13,30)(14,28)(15,17)(16,29)(18,23)(21,36)(24,35)(25,34)(27,31)(32,33);(2,4)(5,6)(7,17)(8,13)(9,14)(10,15)(11,16)(12,26)(18,33)(19,29)(20,28)(21,24)(22,27)(23,31)(25,32)(30,35)(34,36);(5,8,11,25)(6,13,16,32)(7,18,20,35)(9,24,10,23)(12,27,29,36)(14,21,15,31)(17,33,28,30)(19,34,26,22)Polycyclic presentation:
2^9|d^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,g^c=gi,e^d=ei,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 324856);