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 | |
4270 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,8)(3,5,4,9)(6,10,14,15)(11,13,12,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,34)(35,42)(36,39)(37,40)(38,41)(43,48)(44,46)(45,47);(2,8)(3,4)(10,15)(11,12)(19,26)(20,28)(22,23)(24,31)(34,36)(37,42)(41,44)(45,48);(1,3,7,4)(2,9,8,5)(6,11,14,12)(10,16,15,13)(17,19)(18,22)(20,27)(21,26)(23,25)(24,29)(28,32)(30,31)(33,35)(34,37)(36,42)(38,43)(39,40)(41,45)(44,48)(46,47);(1,4,7,3)(2,5,8,9)(6,12,14,11)(10,13,15,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,32)(29,31)(33,36,39,34)(35,42,40,37)(38,41,46,44)(43,45,47,48);(1,5)(2,3)(4,8)(6,13)(7,9)(10,11)(12,15)(14,16)(17,20,21,28)(18,24,25,31)(19,27,26,32)(22,29,23,30)(33,37,39,42)(34,40,36,35)(38,45,46,48)(41,47,44,43);(1,6,7,14)(2,10,8,15)(3,11,4,12)(5,13,9,16)(17,18)(19,22)(20,24)(21,25)(23,26)(27,29)(28,31)(30,32)(33,38,39,46)(34,41,36,44)(35,43,40,47)(37,45,42,48)Polycyclic presentation:
2^9|a^2=ghi,c^2=ghi,d^2=g,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8400030);