Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #94407 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7292 |
| commutator | C2×C16 | C42 |
| Frattini, cosocle | C4×M32 | C22 |
| prime radical | C23 | #365 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C4×C8 ◁ C4×M32 ◁ G | |
| LECS | G ▷ C4×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2)(3,14,16,41,19,34,20,52)(4,21,23,38)(5,26)(6,28)(7,13)(8,15,30,44,32,17,33,47)(9,24,10,37)(11,27)(12,29)(18,50,48,53,43,64,51,42)(22,55)(25,58,46,54)(31,61,60,49,40,63,59,45)(35,56)(36,62,39,57);(1,3,4,17,5,18,24,49,6,20,25,47,27,51,57,63,7,19,23,15,22,43,37,45,29,16,46,44,56,48,62,61)(2,8,10,14,11,31,38,42,12,33,39,41,26,59,58,50,13,32,9,34,35,40,21,53,28,30,36,52,55,60,54,64)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fi,d^2=g,e^2=f,f^2=h,h^2=i,b^a=bc,c^a=cf,e^a=efgi,f^a=fh,h^a=hi,c^b=cg,d^b=dh,g^b=gi,d^c=di,e^c=ei,e^d=ei50 conjugacy classes — view the full character table.
| maximal subgroup of | #72287603, #72287606, #72287610, … |
| maximal normal subgroup of | #72287603, #72287606, #72287610, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1603);