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 | |
4174 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,8)(3,9,11,6)(4,7,13,12)(10,15,16,14)(17,18,23,20)(19,24,30,27)(21,31,29,25)(22,26,28,32)(33,34,40,44)(35,41,38,39)(36,46,45,42)(37,43,47,48);(1,2)(3,6)(4,7)(5,8)(9,11)(10,14)(12,13)(15,16)(17,19)(18,24)(20,27)(21,28)(22,29)(23,30)(25,26)(31,32)(33,35,40,38)(34,41,44,39)(36,47,45,37)(42,43,46,48);(17,18,23,20)(19,27,30,24)(21,25,29,31)(22,26,28,32)(33,36)(34,42)(35,37)(38,47)(39,43)(40,45)(41,48)(44,46);(1,2)(3,6)(4,12)(5,8)(7,13)(9,11)(10,15)(14,16)(17,20,23,18)(19,27,30,24)(21,31,29,25)(22,26,28,32)(33,37,40,47)(34,43,44,48)(35,45,38,36)(39,46,41,42);(1,3,5,11)(2,6,8,9)(4,10,13,16)(7,14,12,15)(17,21,23,29)(18,25,20,31)(19,28,30,22)(24,26,27,32)(33,38,40,35)(34,39,44,41)(36,47,45,37)(42,43,46,48);(1,4)(2,7)(3,10)(5,13)(6,14)(8,12)(9,15)(11,16)(17,22,23,28)(18,26,20,32)(19,29,30,21)(24,25,27,31)(33,39)(34,35)(36,43)(37,42)(38,44)(40,41)(45,48)(46,47)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=hi,d^2=i,e^2=gi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8378335);