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 | #41310 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4062 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(4,12)(8,13)(10,16)(14,15)(18,22)(19,25)(24,29)(26,31)(33,34)(35,44)(36,37)(38,45)(39,43)(40,48)(41,47)(42,46);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32)(33,35)(34,40)(36,45)(37,46)(38,47)(39,48)(41,42)(43,44);(2,6)(7,9)(8,13)(14,15)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,36,39,41)(34,37,43,47)(35,45,48,42)(38,40,46,44);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,37,39,47)(34,41,43,36)(35,46,48,38)(40,42,44,45);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32)(33,34)(35,40)(36,37)(38,42)(39,43)(41,47)(44,48)(45,46);(1,4)(2,8)(3,10)(5,12)(6,13)(7,14)(9,15)(11,16)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,38,39,46)(34,42,43,45)(35,47,48,37)(36,40,41,44)Polycyclic presentation:
2^9|b^2=h,c^2=gh,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8062301);