Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418508 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2675 |
| 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 | |
1535 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)(7,16)(8,20)(9,25)(10,27)(11,15)(12,18)(13,23)(14,26)(17,32)(19,30)(21,28)(22,34)(24,35)(29,36)(31,33);(5,7)(6,12)(8,21)(9,10)(11,19)(13,28)(14,30)(15,25)(16,27)(17,24)(18,26)(20,34)(22,36)(23,33)(29,32)(31,35);(1,2,3,4)(5,8,10,23,11,24,26,36)(6,13,14,20,15,29,27,35)(7,17,18,33,19,21,9,22)(12,28,16,31,25,32,30,34);(5,9,26,19,11,18,10,7)(6,12,27,30,15,25,14,16)(8,22,36,21,24,33,23,17)(13,28,35,34,29,32,20,31)Polycyclic presentation:
2^9|c^2=f,d^2=ei,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,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, 330850);