Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7487870 |
| 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 | |
1900 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)(3,14)(4,9)(5,10)(6,11)(7,13)(8,16)(12,21)(15,29)(17,23)(18,22)(19,26)(20,27)(24,31)(25,32)(28,30);(1,3,7,17)(2,8,13,10)(4,15,19,30)(5,21,16,6)(9,24,26,22)(11,14,12,23)(18,32,31,20)(25,28,27,29);(1,4)(2,9)(3,15)(5,18)(6,20)(7,19)(8,22)(10,24)(11,27)(12,25)(13,26)(14,28)(16,31)(17,30)(21,32)(23,29);(1,5,6,3,7,16,21,17)(2,10,11,14,13,8,12,23)(4,18,20,15,19,31,32,30)(9,22,27,29,26,24,25,28);(2,11,13,12)(4,19)(8,23,10,14)(9,25,26,27)(15,30)(18,31)(20,32)(22,28,24,29)Polycyclic presentation:
2^9|b^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,e^c=ei,g^d=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72215721, #72216601, C2×#7487871, … |
| maximal normal subgroup of | #72215721, #72216601, C2×#7487871, … |
| maximal quotient of | #72250788, C2×#7487871, #72256189, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7395067);