Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454664 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2622 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
795 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,15,11,16,10,12)(7,13,19,30,20,28,17,26)(8,21,23,14,24,27,25,32)(18,31,22,35,33,36,34,29);(1,2,3,4)(6,12)(7,17)(9,10)(13,28)(14,27)(15,16)(18,22)(19,20)(21,32)(23,25)(29,35)(33,34);(2,4)(5,7)(6,13)(8,22,24,34)(9,19)(10,17)(11,20)(12,26)(14,31,32,36)(15,30)(16,28)(18,23,33,25)(21,29,27,35);(5,8)(6,14)(7,18)(9,25)(10,23)(11,24)(12,27)(13,29)(15,21)(16,32)(17,22)(19,34)(20,33)(26,31)(28,35)(30,36)Polycyclic presentation:
2^9|a^2=e,b^2=gh,c^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=ei,g^b=gi,h^b=hi,d^c=de,g^c=gi,h^c=hi,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, 379439);