Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23472 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
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×#400 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1618 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)(7,17)(8,13,24,30)(9,16)(10,14)(11,15)(12,19)(18,33,34,29)(20,27)(21,28)(22,36,35,26)(23,32,25,31);(2,4)(5,7,11,19,10,21,9,20)(6,12,15,28,14,27,16,17)(8,22,25,34,24,35,23,18)(13,29,31,26,30,33,32,36);(2,4)(5,8,11,25,10,24,9,23)(6,13,15,31,14,30,16,32)(7,18,19,22,21,34,20,35)(12,26,28,33,27,36,17,29);(6,14)(12,27)(13,30)(15,16)(17,28)(26,36)(29,33)(31,32)Polycyclic presentation:
2^9|a^2=h,b^2=ef,c^2=ef,e^2=i,g^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 | #72237182, #72237184, #72237187, … |
| maximal normal subgroup of | #72237182, #72237184, #72237187, … |
| maximal quotient of | #72239104, #72265693 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265765);