Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6434130 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2631 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1967 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,7)(3,11)(4,16)(5,17)(8,21)(9,25)(10,20)(12,24)(13,29)(18,19)(27,30)(28,31);(1,2)(3,12)(4,9)(5,20)(6,10)(7,17)(8,15)(11,21)(13,28)(14,24)(16,26)(18,22)(19,25)(23,30)(27,31)(29,32);(1,3)(2,8)(4,13)(5,14)(6,11)(7,21)(9,23)(10,24)(12,20)(15,17)(16,29)(18,30)(19,27)(22,31)(25,32)(26,28);(1,4,17,18,6,19,5,16)(2,9,7,22,10,25,20,26)(3,13,15,30,11,27,14,29)(8,23,21,31,24,32,12,28);(2,10)(7,20)(8,24)(9,25)(12,21)(22,26)(23,32)(28,31)Polycyclic presentation:
2^9|d^2=fg,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,h^a=hi,c^b=ch,e^b=ei,f^b=fi,g^b=gi,h^b=hi,h^c=hi,f^d=fi,g^d=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72215721, #72215972, #72216037, … |
| maximal normal subgroup of | #72215721, #72215972, #72216037, … |
| maximal quotient of | #72224750, #72224752, #72224753, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6341326);