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 | #57995 | 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:
(1,2,6,3)(4,7,11,9)(5,8,14,10)(12,15,13,16)(17,18)(19,23)(20,21)(22,24)(25,29)(26,27)(28,30)(31,32)(33,34)(35,37)(36,44)(38,46)(39,40)(41,48)(42,43)(45,47);(2,3)(4,11)(8,10)(12,13)(17,19,22,28)(18,23,24,30)(20,26,25,31)(21,27,29,32)(33,35)(34,37)(36,45)(38,43)(39,41)(40,48)(42,46)(44,47);(18,24)(20,25)(23,30)(26,31)(34,40)(35,41)(36,46)(43,47);(1,3,6,2)(4,9,11,7)(5,10,14,8)(12,16,13,15)(17,20,22,25)(18,21,24,29)(19,26,28,31)(23,27,30,32)(33,36)(34,38)(35,42)(37,47)(39,46)(40,44)(41,45)(43,48);(1,4)(2,7)(3,9)(5,13)(6,11)(8,16)(10,15)(12,14)(17,19)(18,23)(20,26)(21,27)(22,28)(24,30)(25,31)(29,32)(33,37)(34,35)(36,47)(38,42)(39,48)(40,41)(43,46)(44,45);(1,5,6,14)(2,8,3,10)(4,12,11,13)(7,15,9,16)(17,21)(18,25)(19,27)(20,24)(22,29)(23,31)(26,30)(28,32)(33,38)(34,36)(35,43)(37,45)(39,44)(40,46)(41,47)(42,48)Polycyclic presentation:
2^9|a^2=g,b^2=hi,d^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,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, 9854452);