Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419829 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15273 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1614 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)(3,9,13,7)(4,8)(5,18)(6,10)(11,21)(12,22,30,25)(14,23,19,28)(15,32,31,24)(16,29)(17,26)(20,27);(1,3)(2,7)(4,15)(5,19)(6,13)(8,24)(9,10)(11,28)(12,16)(14,21)(17,31)(18,23)(20,30)(22,27)(25,29)(26,32);(1,4)(2,8)(3,12)(5,20)(6,17)(7,22)(9,25)(10,26)(11,29)(13,30)(14,31)(15,19)(16,21)(18,27)(23,32)(24,28);(1,3)(2,9,10,7)(4,12)(5,14)(6,13)(8,25,26,22)(11,28,18,23)(15,16)(17,30)(19,21)(20,31)(24,27,32,29)Polycyclic presentation:
2^9|a^2=f,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^c=hi,f^d=fi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72237692, #72240441, #72240448, … |
| maximal normal subgroup of | #72237692, #72240441, #72240448, … |
| maximal quotient of | #72237695, #72237701 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327025);