Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #38214 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4014 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,12,13,8)(4,14,11,6)(5,10,15,16)(17,18)(19,24)(20,23)(21,25)(22,26)(27,30)(28,31)(29,32)(33,34,39,40)(35,43,38,41)(36,37,42,47)(44,45,46,48);(1,3)(2,8)(4,5)(6,10)(7,13)(9,12)(11,15)(14,16)(17,19,23,26)(18,24,20,22)(21,29,31,27)(25,32,28,30)(34,40)(37,47)(41,43)(45,48);(2,9)(3,13)(4,11)(10,16)(18,20)(19,26)(25,28)(27,29)(33,35,39,38)(34,41,40,43)(36,46,42,44)(37,45,47,48);(1,4,7,11)(2,6,9,14)(3,5,13,15)(8,10,12,16)(17,20,23,18)(19,22,26,24)(21,28,31,25)(27,32,29,30)(33,36,39,42)(34,37,40,47)(35,44,38,46)(41,48,43,45);(1,5)(2,10)(3,11)(4,13)(6,12)(7,15)(8,14)(9,16)(17,21,23,31)(18,25,20,28)(19,27,26,29)(22,32,24,30)(33,37)(34,42)(35,45)(36,40)(38,48)(39,47)(41,44)(43,46);(1,6)(2,11)(3,10)(4,9)(5,12)(7,14)(8,15)(13,16)(17,22)(18,19)(20,26)(21,30)(23,24)(25,29)(27,28)(31,32)(33,38,39,35)(34,43,40,41)(36,46,42,44)(37,45,47,48)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=hi,d^2=h,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9633250);