Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #201918 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#447 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1160 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,10)(3,5,9,6)(4,11)(8,12)(13,16)(14,15)(17,18,22,26,23,19,21,25)(20,27,28,32,29,24,30,31);(1,3,6,2,7,9,5,10)(4,11,13,8,14,15,12,16)(17,19,22,25,23,18,21,26)(20,27,30,31,29,24,28,32);(1,4)(2,8)(3,11)(5,12)(6,13)(7,14)(9,15)(10,16)(17,20)(18,24)(19,27)(21,28)(22,30)(23,29)(25,31)(26,32);(1,5,7,6)(2,9,10,3)(4,12,14,13)(8,15,16,11)(17,21,23,22)(18,25,19,26)(20,28,29,30)(24,31,27,32)Polycyclic presentation:
2^9|a^2=g,b^2=egh,d^2=i,e^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,d^b=dh,f^b=fi,e^c=ei,f^c=fh,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72308593, C2×#202142 |
| maximal normal subgroup of | #72308593, C2×#202142 |
| maximal quotient of | C2×#202142, #72309140, #72309143, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 109338);