Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7487864 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #52004 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1878 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)(4,9)(5,10)(6,11)(7,12)(8,14)(15,29)(16,23)(17,25)(18,22)(19,27)(20,28)(21,24)(26,30)(31,32);(1,3,7,16)(2,8,12,24)(4,14,6,21)(5,18,20,31)(9,23,11,13)(10,26,28,29)(15,17,30,19)(22,25,32,27);(1,4,7,6)(2,9,12,11)(3,14,16,21)(5,17,20,19)(8,13,24,23)(10,27,28,25)(15,18,30,31)(22,26,32,29);(1,5,6,19,7,20,4,17)(2,10,11,27,12,28,9,25)(3,15,14,18,16,30,21,31)(8,22,23,26,24,32,13,29);(2,11,12,9)(8,23,24,13)(10,27,28,25)(22,26,32,29)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^2=fh,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=ef,f^a=fi,h^a=hi,c^b=ch,d^b=dfh,e^b=eh,f^b=fi,g^b=gi,d^c=dh,g^d=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72231244, #72231251, #72231252, … |
| maximal normal subgroup of | #72231244, #72231251, #72231252, … |
| maximal quotient of | #72265640, C2×#7487867, #72279407, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7395063);