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)(5,6,10,14,12,15,11,29)(7,16,20,13,21,32,19,31)(8,22,25,27,26,9,24,28)(17,33,23,30,35,18,34,36);(5,7,10,19,12,21,11,20)(6,13,29,32,15,31,14,16)(8,23,24,35,26,34,25,17)(9,18,28,30,22,33,27,36);(5,8,10,24,12,26,11,25)(6,9,14,27,15,22,29,28)(7,17,19,23,21,35,20,34)(13,30,16,18,31,36,32,33);(2,4)(5,9)(6,8)(7,18)(10,28)(11,27)(12,22)(13,23)(14,25)(15,26)(16,17)(19,30)(20,36)(21,33)(24,29)(31,34)(32,35)Polycyclic presentation:
2^9|a^2=g,b^2=ef,c^2=e,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, 367905);