Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6439035 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4983 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(6,11)(7,16)(9,12)(13,28)(14,26)(15,27)(18,31)(19,20)(21,30)(22,23)(29,36)(33,34);(1,2,3,4)(5,6,12,27,10,15,9,11)(7,17,16,13,19,25,20,28)(8,21,23,14,24,26,22,30)(18,29,34,32,33,36,31,35);(5,7,9,16,10,19,12,20)(6,13,11,25,15,28,27,17)(8,18,23,31,24,33,22,34)(14,29,26,35,30,36,21,32);(5,8)(6,14)(7,18)(9,23)(10,24)(11,26)(12,22)(13,29)(15,30)(16,31)(17,32)(19,33)(20,34)(21,27)(25,35)(28,36);(7,19)(13,28)(16,20)(17,25)(18,33)(29,36)(31,34)(32,35)Polycyclic presentation:
2^9|b^2=fi,c^2=g,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,h^a=hi,c^b=ch,d^b=dh,e^c=ei,f^c=fi,h^c=hi,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72255901, #72256282, #72256291, … |
| maximal normal subgroup of | #72255901, #72256282, #72256291, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350490);