Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23900 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#386 |
| prime radical | #2847 | C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1394 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)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(17,18,21,25,22,19,23,27)(20,24,29,31,28,26,30,32);(1,2)(3,6)(4,7)(5,8)(9,11)(10,13)(12,14)(15,16)(17,19,21,27,22,18,23,25)(20,26,29,32,28,24,30,31);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,20)(18,24)(19,26)(21,29)(22,28)(23,30)(25,31)(27,32);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(20,28)(24,26)(29,30)(31,32)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,f^b=fh,d^c=di,e^c=eh,g^c=gh92 conjugacy classes — view the full character table.
| maximal subgroup of | #72268400, #72268404, #72268407, … |
| maximal normal subgroup of | #72268400, #72268404, #72268407, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 101096);