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 | |
4216 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,5,9,14)(4,8,12,15)(10,13,16,11)(17,18)(19,25)(20,24)(21,30)(22,26)(23,29)(27,28)(31,32)(34,39)(35,42)(37,45)(44,47);(1,3)(2,5)(4,10)(6,9)(7,14)(8,13)(11,15)(12,16)(17,19)(18,23)(20,27)(21,28)(22,29)(24,31)(25,26)(30,32)(33,34,38,39)(35,36,42,40)(37,41,45,46)(43,44,48,47);(2,7)(3,9)(8,15)(10,16)(17,20,22,30)(18,21,26,24)(19,27,29,32)(23,28,25,31)(33,35,38,42)(34,40,39,36)(37,43,45,48)(41,47,46,44);(1,2)(3,5)(4,8)(6,7)(9,14)(10,13)(11,16)(12,15)(17,20)(18,24)(19,27)(21,26)(22,30)(23,31)(25,28)(29,32)(33,35)(34,36)(37,43)(38,42)(39,40)(41,44)(45,48)(46,47);(1,4,6,12)(2,8,7,15)(3,10,9,16)(5,13,14,11)(17,19)(18,25)(20,27)(21,31)(22,29)(23,26)(24,28)(30,32)(33,36)(34,35)(37,44)(38,40)(39,42)(41,43)(45,47)(46,48);(1,5)(2,3)(4,11)(6,14)(7,9)(8,16)(10,15)(12,13)(17,21,22,24)(18,20,26,30)(19,28,29,31)(23,27,25,32)(33,37,38,45)(34,41,39,46)(35,43,42,48)(36,44,40,47)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=g,e^2=gh,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^e=fgi77 conjugacy classes — view the full character table.
| maximal normal subgroup of | #72219279, #72219283, #72219285 |
| maximal quotient of | #72219273 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9820798);