Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394382 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2424 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3020 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2096 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,17)(8,10)(9,22)(11,15)(12,16)(13,19)(14,24)(18,32)(20,28)(21,27)(23,30)(25,29)(26,35)(31,33)(34,36);(5,7)(6,13)(8,17)(9,18)(10,20)(11,19)(12,21)(14,26)(15,27)(16,28)(22,31)(23,34)(24,33)(25,35)(29,36)(30,32);(2,4)(5,8,10,6,12,15,11,16)(7,17,19,28,21,27,20,13)(9,22,23,14,25,29,24,30)(18,31,33,32,35,36,34,26);(5,9,12,25)(6,14,16,30)(7,18,21,35)(8,22,15,29)(10,23,11,24)(13,26,28,32)(17,31,27,36)(19,33,20,34)Polycyclic presentation:
2^9|c^2=e,d^2=i,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^c=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301880);