Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417728 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2424 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3020 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2309 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,12,17)(7,18,23,32)(8,24,28,30)(9,27,14,26)(10,16,11,15)(13,22,33,21)(19,31,36,20)(25,29,35,34);(5,7)(6,13)(8,25)(9,29)(10,22)(11,21)(12,23)(14,34)(15,32)(16,18)(17,33)(19,26)(20,24)(27,36)(28,35)(30,31);(2,4)(5,8)(6,14)(7,19)(9,17)(10,26)(11,27)(12,28)(13,31)(15,30)(16,24)(18,34)(20,33)(21,25)(22,35)(23,36)(29,32);(5,9,12,14)(6,8,17,28)(7,20,23,31)(10,30,11,24)(13,19,33,36)(15,27,16,26)(18,35,32,25)(21,34,22,29)Polycyclic presentation:
2^9|a^2=i,d^2=i,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,f^c=fi,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, 324927);