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 | #50320 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3899 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,5)(6,14)(8,10)(9,12)(17,18,22,25)(19,20,27,23)(21,29,30,24)(26,31,32,28)(33,34,39,44)(35,42,37,40)(36,43,38,41)(45,47,46,48);(1,2,7,4)(3,10,5,8)(6,12,14,9)(11,15,13,16)(17,19,22,27)(18,23,25,20)(21,26,30,32)(24,31,29,28)(33,35,39,37)(34,40,44,42)(36,45,38,46)(41,47,43,48);(1,3,7,5)(2,8,4,10)(6,11,14,13)(9,15,12,16)(17,18)(19,20)(21,24)(22,25)(23,27)(26,28)(29,30)(31,32)(33,36,39,38)(34,41,44,43)(35,45,37,46)(40,47,42,48);(1,4,7,2)(3,8,5,10)(6,12,14,9)(11,15,13,16)(17,20)(18,19)(21,28)(22,23)(24,26)(25,27)(29,32)(30,31)(33,37,39,35)(34,42,44,40)(36,45,38,46)(41,47,43,48);(1,5,7,3)(2,8,4,10)(6,13,14,11)(9,15,12,16)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,31)(30,32)(33,34)(35,40)(36,41)(37,42)(38,43)(39,44)(45,47)(46,48);(1,6)(2,9)(3,11)(4,12)(5,13)(7,14)(8,15)(10,16)(17,21,22,30)(18,24,25,29)(19,26,27,32)(20,28,23,31)(33,38,39,36)(34,43,44,41)(35,45,37,46)(40,47,42,48)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=gi,d^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8662177);