Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #456670 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1117 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(5,6,10,16,11,15,12,7)(8,17,20,27,21,31,22,13)(9,18,23,14,24,30,25,29)(19,26,34,32,35,28,36,33);(5,7,10,15,11,16,12,6)(8,17,20,13,21,31,22,27)(9,18,23,29,24,30,25,14)(19,32,34,26,35,33,36,28);(2,4)(5,8,11,21)(6,13,15,27)(7,17,16,31)(9,19,24,35)(10,22,12,20)(14,28,29,26)(18,33,30,32)(23,36,25,34);(5,9)(6,14)(7,18)(8,19)(10,23)(11,24)(12,25)(13,26)(15,29)(16,30)(17,32)(20,34)(21,35)(22,36)(27,28)(31,33)Polycyclic presentation:
2^9|a^2=f,b^2=egh,c^2=i,f^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,e^b=ei,f^b=fi,g^b=gi,d^c=de,f^c=fi,h^c=hi,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 379091);