Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394382 |
| 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 | |
2518 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)(7,19)(8,25)(9,26)(10,18)(11,16)(12,17)(13,22)(14,28)(15,30)(20,31)(21,29)(23,34)(24,27)(32,35)(33,36);(1,2,3,4)(5,7,10,22,12,23,11,24)(6,13,16,19,17,27,18,34)(8,26,28,36,29,20,32,15)(9,21,30,14,31,25,33,35);(5,8)(6,14)(7,20)(9,27)(10,32)(11,28)(12,29)(13,31)(15,24)(16,21)(17,35)(18,25)(19,30)(22,36)(23,26)(33,34);(5,9,12,31)(6,15,17,36)(7,21,23,25)(8,27,29,13)(10,30,11,33)(14,24,35,22)(16,26,18,20)(19,28,34,32)Polycyclic presentation:
2^9|b^2=e,d^2=i,e^2=i,f^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,g^b=gi,e^c=ei,f^c=fi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301579);