Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418863 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2684 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#454 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1945 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,16,11,12,10,15)(7,17,21,26,20,29,18,13)(8,22,23,14,25,30,24,27)(19,28,32,36,34,35,33,31);(1,2)(3,4)(6,12)(7,18)(9,10)(13,17)(19,33)(20,21)(22,30)(23,24)(26,29)(28,31)(32,34)(35,36);(2,4)(5,7)(6,13)(8,19)(9,21)(10,18)(11,20)(12,26)(14,28)(15,29)(16,17)(22,31)(23,32)(24,33)(25,34)(27,35)(30,36);(5,8,11,25)(6,14,12,27)(7,19,20,34)(9,24,10,23)(13,28,26,35)(15,30,16,22)(17,31,29,36)(18,32,21,33)Polycyclic presentation:
2^9|a^2=ef,d^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,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^d=fi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72286161, #72287020, #72287067 |
| maximal normal subgroup of | #72286161, #72287020, #72287067 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326060);