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 | |
4250 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,6,4,8)(5,10,14,16)(11,13,12,15)(17,18,23,20)(19,26,21,24)(22,28,30,25)(27,31,29,32)(33,34,38,40)(35,39,43,46)(36,37,44,45)(41,42,47,48);(1,3,7,4)(2,8,9,6)(5,12,14,11)(10,13,16,15)(17,19,23,21)(18,24,20,26)(22,27,30,29)(25,31,28,32)(33,35,38,43)(34,39,40,46)(36,41,44,47)(37,42,45,48);(2,9)(3,4)(10,16)(11,12)(18,20)(22,30)(24,26)(27,29)(34,40)(36,44)(39,46)(41,47);(1,4,7,3)(2,6,9,8)(5,12,14,11)(10,13,16,15)(17,20,23,18)(19,26,21,24)(22,28,30,25)(27,32,29,31)(33,36)(34,37)(35,41)(38,44)(39,42)(40,45)(43,47)(46,48);(1,5,7,14)(2,10,9,16)(3,11,4,12)(6,13,8,15)(17,21,23,19)(18,24,20,26)(22,29,30,27)(25,31,28,32)(33,35)(34,39)(36,41)(37,42)(38,43)(40,46)(44,47)(45,48);(1,6)(2,3)(4,9)(5,13)(7,8)(10,11)(12,16)(14,15)(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)(33,37)(34,36)(35,42)(38,45)(39,41)(40,44)(43,48)(46,47)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^c=fgi,e^d=egi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9385511);