Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417738 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2018 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1306 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)(7,16)(8,13)(9,12)(10,15)(11,19)(14,25)(17,28)(18,34)(20,27)(21,36)(22,29)(23,31)(24,32)(26,33)(30,35);(5,7)(6,11)(8,21)(9,17)(10,20)(12,16)(13,30)(14,27)(15,28)(18,22)(19,25)(23,35)(24,33)(26,31)(29,36)(32,34);(6,12)(7,17)(9,25)(13,29)(14,15)(16,27)(18,33)(19,20)(21,35)(22,23)(30,34)(31,32);(2,4)(5,8,10,24)(6,13,15,32)(7,18)(9,22,25,23)(11,26)(12,29,14,31)(16,30)(17,33)(19,21)(20,35)(27,34)(28,36)Polycyclic presentation:
2^9|d^2=ef,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,g^d=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 335075);