Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419839 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3041 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1381 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)(7,12)(8,21)(9,11)(10,15)(13,23)(14,17)(16,25)(18,27)(19,28)(20,29)(22,26)(24,30)(31,35)(32,34)(33,36);(1,2)(3,4)(6,11)(7,16)(8,22)(9,17)(13,21)(14,15)(18,32)(19,20)(23,24)(25,28)(27,36)(34,35);(2,4)(5,7,17,19,10,20,9,16)(6,12,11,25,15,29,14,28)(8,18,22,33,23,32,24,31)(13,27,26,35,21,34,30,36);(5,8)(6,13)(7,18)(9,24)(10,23)(11,26)(12,27)(14,30)(15,21)(16,31)(17,22)(19,33)(20,32)(25,35)(28,36)(29,34)Polycyclic presentation:
2^9|c^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,g^b=gi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330368);