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 | |
4253 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,8,12)(3,11,6,7)(4,10,5,9)(13,14,15,16)(17,18)(19,24)(20,29)(21,25)(22,28)(23,27)(26,32)(30,31)(33,34,37,40)(35,38,42,46)(36,39,44,43)(41,45,48,47);(1,3,8,6)(2,7,12,11)(4,13,5,15)(9,14,10,16)(17,19,23,30)(18,24,27,31)(20,26,21,28)(22,29,32,25)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,44)(47,48);(1,4,8,5)(2,9,12,10)(3,13,6,15)(7,14,11,16)(17,20,23,21)(18,25,27,29)(19,28,30,26)(22,24,32,31)(33,35,37,42)(34,38,40,46)(36,41,44,48)(39,45,43,47);(1,5,8,4)(2,10,12,9)(3,13,6,15)(7,14,11,16)(17,21,23,20)(18,25,27,29)(19,26,30,28)(22,24,32,31)(33,34)(35,38)(36,43)(37,40)(39,44)(41,47)(42,46)(45,48);(1,6,8,3)(2,11,12,7)(4,13,5,15)(9,14,10,16)(17,22)(18,26)(19,29)(20,31)(21,24)(23,32)(25,30)(27,28)(33,35)(34,38)(36,41)(37,42)(39,45)(40,46)(43,47)(44,48);(1,7,8,11)(2,3,12,6)(4,14,5,16)(9,13,10,15)(17,18)(19,24)(20,25)(21,29)(22,26)(23,27)(28,32)(30,31)(33,36,37,44)(34,39,40,43)(35,41,42,48)(38,45,46,47)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=gi,d^2=h,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8442554);