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 | #52846 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3920 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,5)(3,11,14,8)(4,15,13,9)(6,16,12,10)(17,18)(19,25)(20,27)(21,29)(22,24)(23,30)(26,28)(31,32)(33,34,39,42)(35,37,41,38)(36,44,47,40)(43,46,48,45);(1,3)(2,8)(4,12)(5,11)(6,13)(7,14)(9,10)(15,16)(17,19)(18,20)(21,26)(22,27)(23,28)(24,25)(29,31)(30,32)(33,35,39,41)(34,38,42,37)(36,43,47,48)(40,46,44,45);(1,2,7,5)(3,8,14,11)(4,15,13,9)(6,10,12,16)(17,18,22,24)(19,20,27,25)(21,23,30,29)(26,28,32,31)(33,35)(34,37)(36,43)(38,42)(39,41)(40,45)(44,46)(47,48);(1,4,7,13)(2,9,5,15)(3,12,14,6)(8,10,11,16)(17,20,22,25)(18,19,24,27)(21,28,30,31)(23,26,29,32)(33,36,39,47)(34,40,42,44)(35,43,41,48)(37,45,38,46);(1,5,7,2)(3,11,14,8)(4,9,13,15)(6,16,12,10)(17,19)(18,20)(21,26)(22,27)(23,28)(24,25)(29,31)(30,32)(33,37,39,38)(34,41,42,35)(36,45,47,46)(40,48,44,43);(1,6,7,12)(2,10,5,16)(3,13,14,4)(8,9,11,15)(17,21)(18,23)(19,26)(20,28)(22,30)(24,29)(25,31)(27,32)(33,38,39,37)(34,35,42,41)(36,46,47,45)(40,43,44,48)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=gi,d^2=g,e^2=gh,f^2=gh,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, 8066302);