Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393986 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14715 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4036 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,13)(4,10)(5,11)(6,9)(7,12)(8,15)(14,28)(16,23)(17,24)(18,26)(19,22)(20,25)(21,29)(27,31)(30,32);(1,3)(2,8)(4,14)(5,19)(6,15)(7,16)(9,23)(10,21)(11,13)(12,22)(17,30)(18,31)(20,29)(24,28)(25,27)(26,32);(2,9)(5,6)(8,13)(10,24)(11,12)(15,19)(17,20)(21,27)(22,23)(25,26)(28,32)(29,30);(1,4)(2,10)(3,14)(5,20)(6,17)(7,18)(8,21)(9,24)(11,25)(12,26)(13,27)(15,30)(16,31)(19,29)(22,32)(23,28)Polycyclic presentation:
2^9|e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,h^b=hi,e^c=ei,f^c=fi,e^d=ei65 conjugacy classes — view the full character table.
| maximal subgroup of | #72233773, #72233775, C2×#393986, … |
| maximal normal subgroup of | #72233773, #72233775, C2×#393986, … |
| maximal quotient of | C2×#393986, #72236820, #72236822, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301182);