Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23472 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#124 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#394 |
| prime radical | C22×#575 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4330 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)(5,6,10,15,11,14,9,25)(7,16,18,12,19,29,26,28)(8,20,22,31,24,13,23,32)(17,33,21,36,35,30,34,27);(5,7)(6,12)(8,21)(9,26)(10,18)(11,19)(13,30)(14,28)(15,29)(16,25)(17,23)(20,33)(22,35)(24,34)(27,32)(31,36);(2,4)(5,8)(6,13)(7,17)(9,22)(10,23)(11,24)(12,27)(14,20)(15,31)(16,33)(18,34)(19,35)(21,26)(25,32)(28,36)(29,30);(6,14)(12,28)(13,20)(15,25)(16,29)(27,36)(30,33)(31,32)Polycyclic presentation:
2^9|a^2=h,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,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=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72239600, #72239611, #72239648, … |
| maximal normal subgroup of | #72239600, #72239611, #72239648, … |
| maximal quotient of | #72226731, #72239660, #72265669, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264722);