Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418458 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2424 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3068 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1562 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,18)(8,13)(9,12)(10,16)(11,17)(14,27)(15,26)(19,30)(20,29)(21,31)(22,28)(23,34)(24,32)(25,33)(35,36);(5,7)(6,12)(8,21)(9,26)(10,27)(11,20)(13,30)(14,17)(15,29)(16,18)(19,24)(22,25)(23,35)(28,32)(31,34)(33,36);(1,2,3,4)(5,8,11,25)(6,13,17,33)(7,19,20,35)(9,22,27,21)(10,23,26,24)(12,28,14,31)(15,32,16,34)(18,30,29,36);(5,9,26,7,11,27,10,20)(6,14,16,18,17,12,15,29)(8,22,24,19,25,21,23,35)(13,31,34,30,33,28,32,36)Polycyclic presentation:
2^9|c^2=fi,d^2=ei,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,g^c=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330808);