Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419839 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1199 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,16,12,17,11,30)(7,18,20,33,22,36,21,13)(8,23,25,32,27,14,26,34)(9,28,24,31,19,15,29,35);(1,2)(3,4)(5,7)(6,13)(8,24)(9,25)(10,21)(11,20)(12,22)(14,15)(16,36)(17,33)(18,30)(19,26)(23,28)(27,29)(31,34)(32,35);(2,4)(5,8,12,27)(6,14,17,23)(7,19,22,9)(10,26,11,25)(13,31,33,35)(15,18,28,36)(16,32,30,34)(20,24,21,29);(5,9)(6,15,17,28)(7,8)(10,29)(11,24)(12,19)(13,32,33,34)(14,18,23,36)(16,31,30,35)(20,26)(21,25)(22,27)Polycyclic presentation:
2^9|a^2=e,c^2=i,d^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,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, 334639);