Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418535 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1727 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)(7,17)(8,22)(9,12)(10,15)(11,16)(13,18)(14,23)(19,28)(20,29)(21,26)(24,30)(25,27)(31,34)(32,35)(33,36);(1,2)(3,4)(6,12)(7,18)(8,23)(9,10)(15,16)(17,26)(19,33)(20,21)(22,27)(24,25)(28,31)(35,36);(2,4)(5,7)(6,13)(8,19)(9,18)(10,20)(11,21)(12,26)(14,28)(15,17)(16,29)(22,31)(23,32)(24,34)(25,33)(27,35)(30,36);(5,8,9,23,11,25,10,24)(6,14,15,22,16,30,12,27)(7,19,18,32,21,33,20,34)(13,28,17,31,29,36,26,35)Polycyclic presentation:
2^9|d^2=e,e^2=i,f^2=i,g^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,f^b=fi,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^c=hi,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325731);