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 | #40999 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3991 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)(7,14)(10,16)(13,15)(17,18)(19,26)(20,24)(21,23)(22,27)(25,30)(28,32)(29,31)(33,34,39,42)(35,36,37,40)(38,46,47,41)(43,48,45,44);(1,2)(3,6)(4,7)(5,8)(9,11)(10,13)(12,14)(15,16)(17,19,22,21)(18,23,27,26)(20,28,30,29)(24,31,25,32)(33,35,39,37)(34,40,42,36)(38,43,47,45)(41,48,46,44);(1,2,5,8)(3,6,11,9)(4,7,12,14)(10,13,16,15)(17,20,22,30)(18,24,27,25)(19,28,21,29)(23,31,26,32)(33,35,39,37)(34,40,42,36)(38,43,47,45)(41,48,46,44);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,20)(18,25)(19,28)(21,29)(22,30)(23,32)(24,27)(26,31)(33,36,39,40)(34,35,42,37)(38,44,47,48)(41,43,46,45);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,18)(19,23)(20,24)(21,26)(22,27)(25,30)(28,31)(29,32)(33,37,39,35)(34,36,42,40)(38,45,47,43)(41,44,46,48);(1,4)(2,7)(3,10)(5,12)(6,13)(8,14)(9,15)(11,16)(17,21,22,19)(18,26,27,23)(20,29,30,28)(24,32,25,31)(33,38,39,47)(34,41,42,46)(35,43,37,45)(36,44,40,48)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=gi,d^2=h,e^2=h,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, 8064495);