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 | |
1897 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)(8,18)(12,22)(15,27)(16,26)(17,31)(19,28)(20,30)(21,25)(23,29)(24,32);(1,3,7,17)(2,8,13,25)(4,19,16,15)(5,20,23,32)(6,21,22,18)(9,10,26,29)(11,14,12,31)(24,27,30,28);(1,4)(2,9)(3,15)(5,21)(6,20)(7,16)(8,10)(11,30)(12,24)(13,26)(14,28)(17,19)(18,23)(22,32)(25,29)(27,31);(1,5,22,19,7,23,6,15)(2,10,12,28,13,29,11,27)(3,16,21,20,17,4,18,32)(8,24,14,26,25,30,31,9);(2,11,13,12)(3,17)(4,16)(8,14,25,31)(9,24,26,30)(10,27,29,28)(18,21)(20,32)Polycyclic presentation:
2^9|b^2=i,d^2=fh,e^2=hi,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, #72219528, C2×#7487873, … |
| maximal normal subgroup of | #72215721, #72219528, C2×#7487873, … |
| maximal quotient of | #72250789, C2×#7487873, #72256186, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7395069);