Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454372 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2675 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2017 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1159 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,16,11,17,29,15)(7,18,22,32,23,35,21,12)(8,24,20,33,25,31,28,13)(9,14,27,34,19,30,26,36);(1,2)(3,4)(5,7,10,22,11,23,29,21)(6,12,16,18,17,32,15,35)(8,9,20,27,25,19,28,26)(13,30,24,36,33,14,31,34);(2,4)(5,8)(6,13)(7,19)(9,23)(10,28)(11,25)(12,30)(14,32)(15,24)(16,31)(17,33)(18,34)(20,29)(21,26)(22,27)(35,36);(5,9)(6,14)(7,20)(8,21)(10,27)(11,19)(12,31)(13,18)(15,36)(16,34)(17,30)(22,25)(23,28)(24,32)(26,29)(33,35)Polycyclic presentation:
2^9|a^2=f,b^2=f,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,h^c=hi,e^d=ei,g^d=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 363582);