Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418535 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2018 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1280 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,15,11,13,10,16)(7,17,18,29,20,27,21,12)(8,22,24,32,25,14,26,31)(19,33,23,28,34,30,35,36);(1,2)(3,4)(5,7)(6,12)(8,23)(9,21)(10,18)(11,20)(13,29)(14,30)(15,27)(16,17)(19,24)(22,33)(25,35)(26,34)(28,31)(32,36);(2,4)(6,13)(7,18)(9,10)(12,27)(14,22)(17,29)(19,23)(20,21)(24,26)(28,33)(30,36)(34,35);(5,8)(6,14,13,22)(7,19,20,34)(9,24)(10,26)(11,25)(12,28)(15,31,16,32)(17,30)(18,23,21,35)(27,33)(29,36)Polycyclic presentation:
2^9|a^2=e,d^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,e^c=ei,g^c=gi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 334626);