Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #420420 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1721 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)(7,14)(8,13)(9,23,26,15)(10,17)(11,16)(12,18)(19,30,34,35)(20,29)(21,28)(22,27,33,36)(24,31,25,32);(1,2)(3,4)(5,7)(6,13)(8,11)(9,22)(10,20)(12,21)(14,17)(15,30)(16,28)(18,29)(19,24)(23,35)(25,34)(26,33)(27,31)(32,36);(2,4)(5,8,10,21,12,20,11,7)(6,14,16,29,18,28,17,13)(9,22,24,34,26,33,25,19)(15,30,31,36,23,35,32,27);(5,9,10,24,12,26,11,25)(6,15,16,31,18,23,17,32)(7,19,8,22,21,34,20,33)(13,27,14,30,29,36,28,35)Polycyclic presentation:
2^9|a^2=ef,c^2=e,d^2=e,e^2=i,f^2=i,g^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,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,h^c=hi,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327720);