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 | |
4432 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,4)(3,7,10,11)(5,8,13,12)(9,14,15,16)(17,18)(19,26)(20,23)(21,29)(22,24)(25,30)(27,31)(28,32)(33,34,38,39)(35,36,40,44)(37,45,46,41)(42,48,47,43);(2,4)(7,11)(8,12)(14,16)(17,19,22,20)(18,23,24,26)(21,27,30,28)(25,31,29,32)(33,35,38,40)(34,36,39,44)(37,42,46,47)(41,43,45,48);(5,13)(8,12)(9,15)(14,16)(18,24)(21,30)(23,26)(27,28)(34,39)(35,40)(37,46)(43,48);(1,3)(2,7)(4,11)(5,9)(6,10)(8,14)(12,16)(13,15)(17,18)(19,23)(20,26)(21,25)(22,24)(27,31)(28,32)(29,30)(33,34)(35,36)(37,41)(38,39)(40,44)(42,43)(45,46)(47,48);(1,4,6,2)(3,7,10,11)(5,12,13,8)(9,14,15,16)(17,20,22,19)(18,23,24,26)(21,28,30,27)(25,31,29,32)(33,36)(34,40)(35,39)(37,43)(38,44)(41,47)(42,45)(46,48);(1,5,6,13)(2,8,4,12)(3,9,10,15)(7,14,11,16)(17,21,22,30)(18,25,24,29)(19,27,20,28)(23,31,26,32)(33,37)(34,41)(35,42)(36,43)(38,46)(39,45)(40,47)(44,48)Polycyclic presentation:
2^9|a^2=gh,b^2=h,e^2=g,f^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, 9385536);