Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6442596 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#535 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C2 | C2×#2404 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3839 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,12)(4,5)(6,25)(7,11)(8,10)(9,15)(13,29)(14,18)(16,28)(17,27)(19,24)(20,22)(21,30)(23,26)(31,32);(1,3)(2,9)(4,18)(5,21)(6,17)(7,15)(8,16)(10,27)(11,28)(12,25)(13,19)(14,24)(20,32)(22,29)(23,31)(26,30);(1,4)(2,5)(3,13,16,30)(6,26)(7,19)(8,20)(9,14,27,31)(10,22)(11,24)(12,21,28,29)(15,32,17,18)(23,25);(1,5)(2,4)(3,14,16,31)(6,23)(7,24)(8,22)(9,13,27,30)(10,20)(11,19)(12,18,28,32)(15,29,17,21)(25,26);(2,10)(5,22)(9,27)(11,25)(12,28)(14,31)(21,29)(23,24)Polycyclic presentation:
2^9|c^2=g,d^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,h^b=hi,f^c=fi,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72249649, #72261390, #72261457, … |
| maximal normal subgroup of | #72249649, #72261390, #72261457, … |
| maximal quotient of | #72256044, #72262271, #72302401, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6358973);