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 | |
4255 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,10)(3,8,4,6)(5,9,14,16)(11,15,12,13)(18,23)(19,25)(21,29)(28,31)(33,34,39,43)(35,38,45,47)(36,42,46,40)(37,41,44,48);(3,4)(6,8)(11,12)(13,15)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,32)(29,30)(33,35,39,45)(34,38,43,47)(36,37,46,44)(40,48,42,41);(1,3,7,4)(2,8,10,6)(5,12,14,11)(9,13,16,15)(17,19)(18,20)(21,27)(22,25)(23,24)(26,28)(29,32)(30,31)(33,36,39,46)(34,40,43,42)(35,44,45,37)(38,41,47,48);(1,4,7,3)(2,6,10,8)(5,11,14,12)(9,15,16,13)(17,19)(18,24)(20,23)(21,27)(22,25)(26,31)(28,30)(29,32)(33,37)(34,41)(35,36)(38,42)(39,44)(40,47)(43,48)(45,46);(1,5)(2,9)(3,11)(4,12)(6,13)(7,14)(8,15)(10,16)(17,20,22,24)(18,25,23,19)(21,28,29,31)(26,32,30,27)(33,36)(34,42)(35,37)(38,41)(39,46)(40,43)(44,45)(47,48);(1,6)(2,3)(4,10)(5,13)(7,8)(9,11)(12,16)(14,15)(17,21)(18,26)(19,27)(20,28)(22,29)(23,30)(24,31)(25,32)(33,38)(34,35)(36,41)(37,42)(39,47)(40,44)(43,45)(46,48)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=gi,d^2=hi,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10270870);