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 | #54046 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3767 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,7,11,9)(4,12,13,8)(10,15,16,14)(17,18)(19,20)(21,29)(22,24)(23,27)(25,30)(26,31)(28,32)(33,34,39,43)(35,36,46,40)(37,41,38,42)(44,47,45,48);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,19)(18,23)(20,24)(21,26)(22,27)(25,31)(28,29)(30,32)(33,35,39,46)(34,40,43,36)(37,44,38,45)(41,48,42,47);(2,6)(7,9)(8,12)(14,15)(18,24)(20,23)(21,30)(26,32)(33,35)(34,36)(37,44)(38,45)(39,46)(40,43)(41,47)(42,48);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,20,22,23)(18,19,24,27)(21,28,30,31)(25,26,29,32)(33,36)(34,35)(37,47)(38,48)(39,40)(41,44)(42,45)(43,46);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,15)(13,16)(17,19)(18,20)(21,26)(22,27)(23,24)(25,28)(29,31)(30,32)(33,37,39,38)(34,41,43,42)(35,44,46,45)(36,47,40,48);(1,4,5,13)(2,8,6,12)(3,10,11,16)(7,14,9,15)(17,21,22,30)(18,25,24,29)(19,26,27,32)(20,28,23,31)(33,38,39,37)(34,42,43,41)(35,45,46,44)(36,48,40,47)Polycyclic presentation:
2^9|a^2=i,b^2=gh,d^2=gi,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^c=fgi,e^d=egi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9388729);