Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417802 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2669 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3018 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1875 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,16)(8,20)(9,23)(10,15)(11,18)(12,21)(13,19)(14,26)(17,27)(22,32)(24,31)(25,28)(29,35)(30,36)(33,34);(1,2)(3,4)(5,7,9,18,10,19,26,25)(6,11,14,16,15,28,23,13)(8,17,21,34,22,35,24,36)(12,29,20,33,31,27,32,30);(5,8)(6,12)(7,17)(9,24)(10,22)(11,27)(13,30)(14,32)(15,31)(16,33)(18,36)(19,35)(20,23)(21,26)(25,34)(28,29);(2,4)(5,7)(6,13)(8,17)(9,25)(10,19)(11,23)(12,30)(14,28)(15,16)(18,26)(20,27)(21,36)(22,35)(24,34)(29,32)(31,33)Polycyclic presentation:
2^9|b^2=f,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,g^b=gi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327972);