Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454681 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2622 |
| 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 | |
937 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(6,12)(7,17)(8,22)(9,10)(14,30)(15,16)(18,28)(19,33)(20,21)(24,25)(27,31)(35,36);(1,2,3,4)(5,6)(7,18)(8,14)(9,15)(10,12)(11,16)(13,17)(19,31)(20,29)(21,28)(22,32)(23,27)(24,26)(25,30)(33,36)(34,35);(2,4)(5,7)(6,13,16,29)(8,23)(9,17)(10,20)(11,21)(12,18,15,28)(14,31,30,36)(19,24)(22,33)(25,34)(26,35,32,27);(5,8,11,25)(6,14,16,30)(7,19,21,33)(9,22,10,24)(12,26,15,32)(13,27,29,35)(17,23,20,34)(18,31,28,36)Polycyclic presentation:
2^9|b^2=gh,c^2=ef,d^2=i,e^2=i,f^2=i,g^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,g^b=gi,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, 379478);