Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455572 |
| 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×#2008 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1122 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)(7,18)(8,23)(9,17)(10,16)(11,15)(12,20)(13,21)(14,26)(19,28)(22,29)(24,34)(25,33)(27,32)(30,36)(31,35);(1,2)(3,4)(5,7,11,22,10,21,9,20)(6,12,17,13,16,29,15,18)(8,19,25,24,27,35,26,36)(14,28,32,30,33,31,23,34);(2,4)(5,7)(6,13)(8,24)(9,22)(10,21)(11,20)(12,17)(14,31)(15,29)(16,18)(19,25)(23,34)(26,35)(27,36)(28,33)(30,32);(5,8)(6,14)(7,19)(9,26)(10,27)(11,25)(12,28)(13,30)(15,23)(16,33)(17,32)(18,34)(20,36)(21,35)(22,24)(29,31)Polycyclic presentation:
2^9|b^2=ef,e^2=i,g^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,h^b=hi,d^c=de,e^c=ei,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, 362891);