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 |
| prime radical | #43973 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4069 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)(3,8)(4,10)(5,11)(6,13)(7,9)(12,16)(14,15)(17,18,22,23)(19,25,26,20)(21,29,30,24)(27,28,32,31)(33,34)(35,43)(36,39)(37,41)(38,47)(40,45)(42,48)(44,46);(1,3,7,4)(2,8,9,10)(5,13,15,12)(6,14,16,11)(17,19,22,26)(18,20,23,25)(21,27,30,32)(24,28,29,31)(33,35,39,45)(34,40,36,43)(37,38,44,48)(41,42,46,47);(2,9)(3,4)(11,14)(12,13)(18,23)(19,26)(24,29)(27,32)(34,36)(40,43)(41,46)(42,47);(1,4,7,3)(2,10,9,8)(5,12,15,13)(6,11,16,14)(17,18)(19,20)(21,24)(22,23)(25,26)(27,28)(29,30)(31,32)(33,36,39,34)(35,43,45,40)(37,46,44,41)(38,47,48,42);(1,5)(2,11)(3,13)(4,12)(6,8)(7,15)(9,14)(10,16)(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,32)(30,31)(33,37)(34,41)(35,38)(36,46)(39,44)(40,42)(43,47)(45,48);(1,6,7,16)(2,12,9,13)(3,14,4,11)(5,10,15,8)(17,21)(18,24)(19,27)(20,28)(22,30)(23,29)(25,31)(26,32)(33,38)(34,42)(35,44)(36,47)(37,45)(39,48)(40,46)(41,43)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,d^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9721934);