Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394464 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#124 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#397 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1846 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,16,19,31)(8,13)(9,14)(10,25)(11,15)(12,18,27,20)(17,32,33,28)(21,35,34,26)(22,29)(23,30)(24,36);(5,7)(6,12)(8,21)(9,20)(10,18)(11,19)(13,28)(14,31)(15,27)(16,25)(17,24)(22,34)(23,33)(26,30)(29,32)(35,36);(5,8,9,23,11,22,10,24)(6,13,14,30,15,29,25,36)(7,17,18,34,19,33,20,21)(12,26,16,28,27,35,31,32);(2,4)(8,22)(13,29)(17,33)(21,34)(23,24)(26,35)(28,32)(30,36)Polycyclic presentation:
2^9|a^2=f,c^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302976);