Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417859 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2433 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1634 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)(7,15)(8,20)(9,24)(10,14)(11,18)(12,22)(13,26)(16,25)(17,31)(19,30)(21,28)(23,33)(27,32)(29,36)(34,35);(5,7)(6,11)(8,21)(9,25)(10,19)(12,28)(13,30)(14,16)(15,24)(17,22)(18,26)(20,27)(23,35)(29,34)(31,33)(32,36);(5,6)(7,16,19,11)(8,12)(9,13)(10,14)(15,18,30,25)(17,31,32,27)(20,22)(21,34,35,28)(23,29)(24,26)(33,36);(2,4)(5,8,9,22,10,23,26,36)(6,12,13,20,14,29,24,33)(7,17,18,21,19,32,25,35)(11,27,15,28,16,31,30,34)Polycyclic presentation:
2^9|c^2=f,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330819);