Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #420064 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #555 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2537 |
| prime radical | C2×#3067 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1346 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)(6,12,16,15)(7,17,20,25)(14,21,30,29)(19,31,34,35);(5,6)(7,18)(8,21)(9,15)(10,12)(11,16)(13,17)(14,22)(19,33)(20,27)(23,30)(24,29)(25,28)(26,34)(31,32)(35,36);(5,7)(6,13)(8,19)(9,17)(10,25)(11,20)(12,18)(14,26)(15,27)(16,28)(21,32)(22,31)(23,35)(24,34)(29,36)(30,33);(2,4)(5,8,9,22,11,24,10,23)(6,14,12,21,16,30,15,29)(7,19,17,31,20,34,25,35)(13,26,18,32,28,33,27,36)Polycyclic presentation:
2^9|a^2=ef,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,g^b=gi,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327260);