Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454021 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1965 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6)(7,16)(8,20)(9,14)(10,26)(11,15)(12,18)(13,24)(17,31)(19,29)(21,33)(22,25)(23,32)(27,30)(28,35)(34,36);(1,2,4,3)(5,7,9,18)(6,12,14,16)(8,21,23,35)(10,19,11,27)(13,31,22,36)(15,29,26,30)(17,24,34,25)(20,28,32,33);(2,3)(5,8)(6,13)(7,17)(9,23)(10,25)(11,24)(12,28)(14,22)(15,32)(16,33)(18,34)(19,35)(20,26)(21,27)(29,31)(30,36);(1,3,4,2)(5,6)(7,12)(8,22)(9,14)(10,26)(11,15)(13,23)(16,18)(17,33)(19,30)(20,24)(21,36)(25,32)(27,29)(28,34)(31,35)Polycyclic presentation:
2^9|b^2=e,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,g^b=gi,d^c=de,f^c=fi,g^c=gi,h^c=hi,f^d=fi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361617);