Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6431739 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C211 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×#392 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
7139 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)(3,13)(4,10)(5,11)(6,9)(7,12)(8,15)(14,27)(16,23)(17,25)(18,26)(19,24)(20,22)(21,29)(28,32)(30,31);(1,3)(2,8)(4,14)(5,20)(6,15)(7,16)(9,23)(10,21)(11,13)(12,22)(17,28)(18,30)(19,29)(24,32)(25,31)(26,27);(2,9)(5,6)(8,13)(10,24)(11,12)(15,20)(18,19)(21,27)(22,23)(25,26)(29,30)(31,32);(1,4)(2,10)(3,14)(5,18)(6,19)(7,17)(8,21)(9,24)(11,26)(12,25)(13,27)(15,29)(16,28)(20,30)(22,31)(23,32);(4,17)(10,25)(14,28)(18,19)(21,31)(24,26)(27,32)(29,30)Polycyclic presentation:
2^9|f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,h^a=hi,c^b=ch,f^b=fi,f^c=fi,g^c=gi,e^d=ei89 conjugacy classes — view the full character table.
| maximal subgroup of | #72215901, #72216141, #72219683, … |
| maximal normal subgroup of | #72215901, #72216141, #72219683, … |
| maximal quotient of | C2×#6431739, D4×#2008, #72252474, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6338935);