Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418464 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60645 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1486 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,16)(8,9)(10,23)(11,15)(12,19)(13,27)(14,22)(17,18)(20,33)(21,35)(24,29)(25,28)(26,32)(30,34)(31,36);(5,7,10,19,11,20,22,34)(6,12,14,16,15,30,23,33)(8,21,24,32,25,17,13,31)(9,26,27,35,28,36,29,18);(5,8,22,24,11,25,10,13)(6,9,23,27,15,28,14,29)(7,17,34,31,20,21,19,32)(12,26,33,35,30,36,16,18);(2,4)(5,9)(6,13)(7,18)(8,23)(10,29)(11,28)(12,17)(14,25)(15,24)(16,32)(19,36)(20,35)(21,30)(22,27)(26,34)(31,33)Polycyclic presentation:
2^9|b^2=e,c^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,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=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330467);