Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394038 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#392 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2008 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1930 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,17)(8,13)(9,16)(10,15)(11,14)(12,19)(18,33)(20,28)(21,27)(22,36)(23,32)(24,30)(25,31)(26,35)(29,34);(5,7)(6,12)(8,22)(9,21)(10,19)(11,20)(13,29)(14,27)(15,28)(16,17)(18,23)(24,35)(25,34)(26,31)(30,33)(32,36);(5,8)(6,13)(7,18)(9,23)(10,25)(11,24)(12,26)(14,30)(15,31)(16,32)(17,29)(19,22)(20,34)(21,35)(27,36)(28,33);(1,2,3,4)(6,14)(12,27)(13,30)(15,16)(17,28)(26,36)(29,33)(31,32)Polycyclic presentation:
2^9|d^2=hi,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 318908);