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 | |
4199 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,3)(7,9)(8,10)(14,15)(17,18,23,24)(19,27,28,22)(20,29,32,25)(21,31,30,26)(33,34,39,38)(35,40,42,36)(37,45,47,41)(43,44,48,46);(1,2,6,3)(4,7,12,9)(5,8,13,10)(11,14,16,15)(17,19,23,28)(18,22,24,27)(20,30,32,21)(25,26,29,31)(33,35)(34,40)(36,38)(37,43)(39,42)(41,46)(44,45)(47,48);(5,13)(8,10)(11,16)(14,15)(18,24)(22,27)(25,29)(26,31)(34,38)(35,42)(41,45)(43,48);(1,3,6,2)(4,7,12,9)(5,10,13,8)(11,14,16,15)(17,20,23,32)(18,25,24,29)(19,21,28,30)(22,31,27,26)(33,36)(34,35)(37,46)(38,42)(39,40)(41,48)(43,45)(44,47);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,21)(18,26)(19,20)(22,29)(23,30)(24,31)(25,27)(28,32)(33,37,39,47)(34,41,38,45)(35,43,42,48)(36,44,40,46);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,22,23,27)(18,19,24,28)(20,31,32,26)(21,29,30,25)(33,38,39,34)(35,36,42,40)(37,45,47,41)(43,44,48,46)Polycyclic presentation:
2^9|a^2=h,b^2=g,d^2=g,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8229415);