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 | #53992 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4014 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(4,12)(8,13)(10,16)(14,15)(17,18)(19,24)(20,29)(21,31)(22,27)(23,28)(25,30)(26,32)(33,34,39,40)(35,44,45,41)(36,48,42,37)(38,43,46,47);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(19,28)(20,30)(23,24)(25,29)(34,40)(35,45)(36,42)(43,47);(2,6)(7,9)(8,13)(14,15)(17,19,22,28)(18,23,27,24)(20,26,30,31)(21,25,32,29)(33,35)(34,41)(36,46)(37,47)(38,42)(39,45)(40,44)(43,48);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,19)(18,24)(20,26)(21,25)(22,28)(23,27)(29,32)(30,31)(33,36,39,42)(34,37,40,48)(35,46,45,38)(41,47,44,43);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,20)(18,25)(19,26)(21,24)(22,30)(23,32)(27,29)(28,31)(33,37)(34,42)(35,47)(36,40)(38,41)(39,48)(43,45)(44,46);(1,4)(2,8)(3,10)(5,12)(6,13)(7,14)(9,15)(11,16)(17,21)(18,26)(19,25)(20,24)(22,32)(23,30)(27,31)(28,29)(33,38,39,46)(34,43,40,47)(35,42,45,36)(37,41,48,44)Polycyclic presentation:
2^9|a^2=g,c^2=gh,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8017942);