Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417802 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2669 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1507 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)(7,15)(8,19)(9,11)(10,25)(12,17)(13,22)(14,27)(16,28)(18,32)(20,36)(21,26)(23,30)(24,29)(31,33)(34,35);(5,7,10,17,11,18,27,29)(6,12,14,15,9,24,25,32)(8,20,22,34,23,16,26,33)(13,28,19,35,21,36,30,31);(1,2,3,4)(5,8,11,23)(6,13,9,21)(7,16,18,20)(10,26,27,22)(12,28,24,36)(14,30,25,19)(15,31,32,35)(17,34,29,33);(5,9,11,6)(7,15,18,32)(8,21,23,13)(10,14,27,25)(12,29,24,17)(16,31,20,35)(19,26,30,22)(28,33,36,34)Polycyclic presentation:
2^9|b^2=e,c^2=ef,d^2=i,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,g^b=gi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330417);