Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #124 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2398 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1380 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)(5,6,11,16)(7,9,14,27)(8,13)(10,18,25,15)(12,28,26,19)(17,30)(20,29)(21,31)(22,33)(23,34)(24,32)(35,36);(2,4)(5,7)(6,12)(8,20)(9,18)(10,28)(11,14)(13,30)(15,27)(16,26)(17,24)(19,25)(21,34)(22,31)(23,35)(29,33)(32,36);(2,4)(5,8)(6,13)(7,17)(9,21)(10,22)(11,23)(12,29)(14,31)(15,32)(16,34)(18,33)(19,35)(20,28)(24,25)(26,36)(27,30);(5,9,25,12,11,27,10,26)(6,14,15,28,16,7,18,19)(8,21,24,29,23,30,22,36)(13,31,32,20,34,17,33,35)Polycyclic presentation:
2^9|a^2=h,d^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,g^c=gi,h^c=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72239887, #72250610, #72250617, … |
| maximal normal subgroup of | #72239887, #72250610, #72250617, … |
| maximal quotient of | #72239888 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265884);