Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453999 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1775 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)(6,12)(7,17)(9,10)(13,18)(14,30)(15,16)(19,22)(20,21)(23,24)(27,32)(31,36)(34,35);(5,6)(7,18)(8,14)(9,15)(10,12)(11,16)(13,21)(17,26)(19,33)(20,29)(22,31)(23,27)(24,30)(25,32)(28,35)(34,36);(2,4)(5,7,11,21)(6,13,16,18)(8,22,25,34)(9,17,10,20)(12,26,15,29)(14,31,32,36)(19,24,35,23)(27,33,30,28);(5,8)(6,14)(7,19)(9,24)(10,23)(11,25)(12,27)(13,28)(15,30)(16,32)(17,22)(18,33)(20,34)(21,35)(26,31)(29,36)Polycyclic presentation:
2^9|a^2=ef,c^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,e^c=ei,f^c=fi,e^d=ei,f^d=fi,g^d=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361883);