Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454041 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| 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 | |
985 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6)(7,17)(8,22)(9,12)(10,15)(11,16)(13,20)(14,25)(18,33)(19,30)(21,27)(23,31)(24,32)(26,28)(29,35)(34,36);(1,3,4,2)(6,12)(9,10)(13,27)(14,28)(15,16)(17,30)(19,20)(22,32)(23,34)(24,25)(29,36)(31,33);(2,3)(5,7)(6,13)(8,23)(9,20)(10,19)(11,21)(12,27)(14,31)(15,17)(16,30)(18,24)(22,33)(25,35)(26,34)(28,29)(32,36);(5,8)(6,14)(7,18)(9,25)(10,24)(11,26)(12,28)(13,29)(15,22)(16,32)(17,31)(19,34)(20,23)(21,35)(27,36)(30,33)Polycyclic presentation:
2^9|a^2=ef,b^2=ef,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,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=gi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 363940);