Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #456670 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×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 | |
916 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)(5,6,10,15,11,16,12,26)(7,17,22,13,21,32,20,33)(8,9,25,28,14,29,24,27)(18,19,23,35,34,36,31,30);(5,7,12,22,11,21,10,20)(6,13,15,17,16,33,26,32)(8,23,25,18,14,31,24,34)(9,19,27,35,29,36,28,30);(5,8,12,25,11,14,10,24)(6,9,26,28,16,29,15,27)(7,18,22,31,21,34,20,23)(13,30,32,36,33,35,17,19);(2,4)(5,9)(6,14)(7,19)(8,16)(10,28)(11,29)(12,27)(13,31)(15,24)(17,34)(18,32)(20,30)(21,36)(22,35)(23,33)(25,26)Polycyclic presentation:
2^9|a^2=g,b^2=ef,c^2=ei,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,g^b=gi,h^b=hi,d^c=de,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 367906);