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 | #40173 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4182 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(4,11)(8,13)(9,15)(14,16)(18,23)(20,27)(21,29)(26,31)(33,34,40,39)(35,43,38,41)(36,46,42,37)(44,45,47,48);(2,6)(7,12)(8,13)(14,16)(18,23)(19,25)(20,27)(28,32)(33,35,40,38)(34,41,39,43)(36,47,42,44)(37,48,46,45);(1,2)(3,7)(4,8)(5,6)(9,14)(10,12)(11,13)(15,16)(17,18,22,23)(19,21,25,29)(20,24,27,30)(26,28,31,32)(33,36)(34,37)(35,44)(38,47)(39,46)(40,42)(41,45)(43,48);(1,3,5,10)(2,7,6,12)(4,9,11,15)(8,14,13,16)(17,19,22,25)(18,21,23,29)(20,26,27,31)(24,28,30,32)(33,37,40,46)(34,42,39,36)(35,45,38,48)(41,47,43,44);(1,2)(3,7)(4,8)(5,6)(9,14)(10,12)(11,13)(15,16)(17,20)(18,24)(19,26)(21,28)(22,27)(23,30)(25,31)(29,32)(33,38,40,35)(34,43,39,41)(36,47,42,44)(37,48,46,45);(1,4)(2,8)(3,9)(5,11)(6,13)(7,14)(10,15)(12,16)(17,21)(18,25)(19,23)(20,28)(22,29)(24,31)(26,30)(27,32)(33,39,40,34)(35,43,38,41)(36,46,42,37)(44,48,47,45)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=gh,d^2=i,e^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=egi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72239255 |
| maximal normal subgroup of | #72239255 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8016378);