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 | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
994 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)(8,14)(9,15)(10,13)(11,16)(12,18)(19,33)(20,25)(21,27)(22,28)(23,29)(24,30)(26,34)(31,35)(32,36);(1,3,4,2)(5,7)(6,12)(8,19)(9,20)(10,18)(11,21)(13,27)(14,26)(15,17)(16,25)(22,32)(23,34)(24,35)(28,33)(29,31)(30,36);(2,3)(6,13)(7,18)(8,22)(9,10)(12,25)(15,16)(20,21)(23,24)(26,31)(28,29)(32,34)(33,36);(5,8,11,24)(6,14,16,30)(7,19,21,35)(9,23,10,22)(12,26,25,36)(13,28,15,29)(17,31,27,33)(18,32,20,34)Polycyclic presentation:
2^9|a^2=ef,b^2=ef,d^2=i,e^2=i,f^2=i,g^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,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,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, 363943);