Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #456658 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
867 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,15,11,17,9,16)(7,14,20,29,21,35,19,12)(8,22,26,33,27,32,25,13)(18,28,24,30,36,31,23,34);(1,2,3,4)(5,7)(6,12,17,29)(8,23)(9,19)(10,20)(11,21)(13,30,33,34)(14,16,35,15)(18,26)(22,31,32,28)(24,27)(25,36);(2,4)(5,8,11,27)(6,13,17,33)(7,18,21,36)(9,26,10,25)(12,28,29,31)(14,34,35,30)(15,32,16,22)(19,24,20,23);(5,7)(6,14,17,35)(8,24)(9,20)(10,19)(11,21)(12,16,29,15)(13,31,33,28)(18,26)(22,30,32,34)(23,27)(25,36)Polycyclic presentation:
2^9|a^2=f,b^2=egh,c^2=i,d^2=ef,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,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, 380357);