Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394382 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2424 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1207 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6)(7,17)(8,22)(9,15)(10,14)(11,16)(12,19)(13,25)(18,27)(20,29)(21,28)(23,36)(24,31)(26,30)(32,34)(33,35);(1,3,4,2)(5,7)(6,12)(8,23)(9,20)(10,19)(11,21)(13,27)(14,28)(15,17)(16,29)(18,26)(22,32)(24,33)(25,34)(30,35)(31,36);(2,3)(5,8,10,25,11,26,9,24)(6,13,15,22,16,31,14,30)(7,18,20,34,21,23,19,33)(12,27,28,35,29,36,17,32);(5,7)(6,12)(8,18)(9,19)(10,20)(11,21)(13,27)(14,17)(15,28)(16,29)(22,35)(23,26)(24,33)(25,34)(30,32)(31,36)Polycyclic presentation:
2^9|a^2=ef,b^2=ef,c^2=ei,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^c=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 304463);