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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4111 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)(3,6)(4,11)(5,7)(8,12)(9,15)(10,13)(14,16)(17,18,23,25)(19,28,21,24)(20,29,31,26)(22,30,32,27)(37,45)(42,46)(43,48)(44,47);(1,3,5,10)(2,6,7,13)(4,9,12,16)(8,14,11,15)(17,19,23,21)(18,24,25,28)(20,30,31,27)(22,26,32,29)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,48)(45,46);(2,7)(6,13)(8,11)(14,15)(18,25)(19,21)(26,29)(27,30)(34,39)(36,40)(42,46)(44,47);(1,3)(2,6)(4,9)(5,10)(7,13)(8,14)(11,15)(12,16)(17,20)(18,26)(19,27)(21,30)(22,24)(23,31)(25,29)(28,32)(33,35,38,41)(34,40,39,36)(37,43,45,48)(42,47,46,44);(1,2)(3,6)(4,8)(5,7)(9,14)(10,13)(11,12)(15,16)(17,21,23,19)(18,24,25,28)(20,30,31,27)(22,29,32,26)(33,36,38,40)(34,41,39,35)(37,44,45,47)(42,48,46,43);(1,4,5,12)(2,8,7,11)(3,9,10,16)(6,14,13,15)(17,22)(18,27)(19,26)(20,24)(21,29)(23,32)(25,30)(28,31)(33,37)(34,42)(35,43)(36,44)(38,45)(39,46)(40,47)(41,48)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=gi,e^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8002913);