Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418976 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2408 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3010 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1872 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,18,12,17,11,16)(7,19,21,13,9,29,22,15)(8,23,26,33,28,14,27,34)(20,35,36,30,25,31,24,32);(1,2)(3,4)(5,7,11,22,12,9,10,21)(6,13,18,29,17,15,16,19)(8,24,27,25,28,36,26,20)(14,31,34,32,23,35,33,30);(2,4)(5,8)(6,14)(7,20)(9,25)(10,27)(11,26)(12,28)(13,30)(15,32)(16,34)(17,23)(18,33)(19,31)(21,24)(22,36)(29,35);(5,9,12,7)(6,15,17,13)(8,25,28,20)(10,21,11,22)(14,32,23,30)(16,19,18,29)(24,26,36,27)(31,33,35,34)Polycyclic presentation:
2^9|a^2=ef,b^2=ei,d^2=i,e^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,c^b=cg,d^b=de,e^c=ei,g^c=gi,e^d=ei,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, 326379);