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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3971 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)(3,4)(5,8)(6,9)(7,11)(10,12)(13,14)(15,16)(17,18,22,25)(19,26,23,20)(21,24,30,32)(27,31,29,28)(35,42)(39,46)(43,47)(44,48);(3,7)(4,11)(10,15)(12,16)(17,19)(18,20)(21,27)(22,23)(24,28)(25,26)(29,30)(31,32)(33,34,38,41)(35,39,42,46)(36,37,40,45)(43,44,47,48);(1,3,6,7)(2,4,9,11)(5,10,14,15)(8,12,13,16)(17,18,22,25)(19,20,23,26)(21,24,30,32)(27,28,29,31)(33,35)(34,39)(36,43)(37,44)(38,42)(40,47)(41,46)(45,48);(1,2)(3,4)(5,13)(6,9)(7,11)(8,14)(10,16)(12,15)(17,19)(18,20)(21,29)(22,23)(24,31)(25,26)(27,30)(28,32)(33,36)(34,37)(35,43)(38,40)(39,44)(41,45)(42,47)(46,48);(1,4)(2,7)(3,9)(5,12)(6,11)(8,15)(10,13)(14,16)(17,20)(18,23)(19,25)(21,28)(22,26)(24,29)(27,32)(30,31)(33,37)(34,40)(35,44)(36,41)(38,45)(39,47)(42,48)(43,46);(1,5)(2,8)(3,10)(4,12)(6,14)(7,15)(9,13)(11,16)(17,21,22,30)(18,24,25,32)(19,27,23,29)(20,28,26,31)(33,35)(34,39)(36,43)(37,44)(38,42)(40,47)(41,46)(45,48)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10330202);