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 | #57146 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4198 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,9)(3,7,10,6)(4,12,13,8)(11,14,16,15)(17,18,22,23)(19,20,27,24)(21,29,30,25)(26,31,32,28)(33,34,38,39)(35,37,42,40)(36,44,46,41)(43,47,48,45);(3,10)(6,7)(11,16)(14,15)(18,23)(20,24)(25,29)(28,31)(34,39)(35,42)(41,44)(43,48);(1,3,5,10)(2,6,9,7)(4,11,13,16)(8,14,12,15)(17,19,22,27)(18,24,23,20)(21,26,30,32)(25,31,29,28)(33,35,38,42)(34,40,39,37)(36,43,46,48)(41,47,44,45);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,20,22,24)(18,19,23,27)(21,28,30,31)(25,26,29,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,46)(45,48);(1,2)(3,7)(4,8)(5,9)(6,10)(11,15)(12,13)(14,16)(17,19)(18,24)(20,23)(21,26)(22,27)(25,31)(28,29)(30,32)(33,36,38,46)(34,41,39,44)(35,43,42,48)(37,45,40,47);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,12)(10,16)(17,21,22,30)(18,25,23,29)(19,26,27,32)(20,28,24,31)(33,37,38,40)(34,42,39,35)(36,45,46,47)(41,48,44,43)Polycyclic presentation:
2^9|a^2=h,c^2=h,d^2=hi,e^2=ghi,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^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8038384);