Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511819 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #32898 |
| 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 | |
2370 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)(4,15)(5,11)(6,20)(7,12)(8,14)(10,13)(16,29)(17,25)(18,24)(19,22)(21,27)(23,31)(26,28)(30,32);(1,3,7,6)(2,8,12,24)(4,13,17,21)(5,14,19,18)(9,22,20,11)(10,23,27,29)(15,26,25,32)(16,28,31,30);(1,3,7,6)(2,9,12,20)(4,13,17,21)(5,18,19,14)(8,22,24,11)(10,25,27,15)(16,30,31,28)(23,32,29,26);(1,4,7,17)(2,10,12,27)(3,13,6,21)(5,16,19,31)(8,23,24,29)(9,25,20,15)(11,26,22,32)(14,28,18,30)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,g^a=gi,h^a=hi,f^b=fhi,e^c=eh,g^d=gi,f^e=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72236997, #72237540, #72261432, … |
| maximal normal subgroup of | #72236997, #72237540, #72261432, … |
| maximal quotient of | #72281312, #72281509, #72281765, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419020);