Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455962 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2826 |
| 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 | |
688 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,14,10,12,23,27)(7,15,18,25,19,30,16,11)(8,13,20,29,22,26,21,28)(17,31,33,35,34,36,32,24);(1,2,3,4)(5,7)(6,11)(8,17)(9,16)(10,19)(12,25)(13,24)(14,30)(15,27)(18,23)(20,32)(21,33)(22,34)(26,35)(28,31)(29,36);(2,4)(6,12)(7,16)(8,20)(9,23)(11,15)(13,28)(18,19)(21,22)(24,35)(25,30)(26,29)(32,33);(5,8,9,20,10,22,23,21)(6,13,27,28,12,26,14,29)(7,17,16,32,19,34,18,33)(11,24,15,31,25,35,30,36)Polycyclic presentation:
2^9|a^2=e,b^2=egh,d^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,d^c=de,e^c=ei,g^c=gi,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, 380151);