Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417661 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60426 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1722 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(5,6)(7,11,18,29)(8,13,21,12)(9,15)(10,24)(14,23)(16,27)(17,31)(19,34,25,30)(20,28)(22,32,26,33)(35,36);(5,7)(6,11)(8,20)(9,18)(10,25)(12,31)(13,28)(14,34)(15,29)(16,22)(17,21)(19,23)(24,30)(26,35)(27,32)(33,36);(2,4)(5,8)(6,12,15,13)(7,16)(9,21)(10,26)(11,27,29,36)(14,33,24,32)(17,19)(18,35)(20,25)(22,23)(28,34,31,30);(5,8)(6,13,15,12)(7,17)(9,21)(10,22)(11,28,29,31)(14,33,24,32)(16,19)(18,20)(23,26)(25,35)(27,34,36,30)Polycyclic presentation:
2^9|a^2=ef,c^2=ei,d^2=ei,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,f^b=fi,g^b=gi,h^b=hi,f^c=fi,g^c=gi,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, 327725);