Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499885 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #23900 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×2−1+4 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1846 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)(4,11)(5,19)(6,23)(7,12)(8,16)(9,18)(10,21)(14,30)(15,25)(17,29)(20,27)(22,28)(24,32)(26,31);(1,3,7,17)(2,8,12,27)(4,15,6,14)(5,20,21,16)(9,25,28,30)(10,29,19,13)(11,24,23,26)(18,31,22,32);(3,14)(4,6)(5,21)(8,24)(9,28)(11,23)(13,25)(15,17)(16,32)(20,31)(26,27)(29,30);(1,4,7,6)(2,9,12,28)(3,15,17,14)(5,22,21,18)(8,25,27,30)(10,23,19,11)(13,24,29,26)(16,31,20,32)Polycyclic presentation:
2^9|b^2=i,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fh,g^b=gi,d^c=di,e^c=eh,g^c=gi,e^d=ei,f^d=fhi,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72236743, #72236949, #72236997, … |
| maximal normal subgroup of | #72236949, #72236997, #72237525, … |
| maximal quotient of | #72311964, #72312154, #72312679, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407085);