Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #422715 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2428 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1885 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)(5,6,12,16)(7,11,15,10)(8,17,22,28)(9,18,26,30)(13,21,29,20)(14,25,23,24)(19,32,36,27)(31,34,33,35);(1,2,3,4)(5,7,10,6,12,15,11,16)(8,13,20,17,22,29,21,28)(9,23,24,30,26,14,25,18)(19,33,34,32,36,31,35,27);(5,8,11,20,12,22,10,21)(6,13,7,17,16,29,15,28)(9,19,24,35,26,36,25,34)(14,27,18,31,23,32,30,33);(2,4)(5,9)(6,14)(7,18)(8,19)(10,25)(11,24)(12,26)(13,27)(15,30)(16,23)(17,31)(20,35)(21,34)(22,36)(28,33)(29,32)Polycyclic presentation:
2^9|a^2=i,b^2=e,c^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^c=ei,g^c=gi,e^d=ei,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330093);