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 | |
4000 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,4)(3,10,9,6)(5,8,14,12)(11,16,15,13)(17,18)(19,23)(20,24)(21,25)(22,26)(27,31)(28,29)(30,32)(33,34,39,42)(35,43,46,37)(36,41,38,40)(44,47,45,48);(1,3)(2,6)(4,10)(5,11)(7,9)(8,13)(12,16)(14,15)(17,18,22,26)(19,23,24,20)(21,25,30,32)(27,31,29,28)(33,35,39,46)(34,37,42,43)(36,44,38,45)(40,47,41,48);(2,4)(6,10)(8,12)(13,16)(17,19,22,24)(18,23,26,20)(21,29,30,27)(25,28,32,31)(33,35)(34,37)(36,44)(38,45)(39,46)(40,47)(41,48)(42,43);(1,4,7,2)(3,10,9,6)(5,8,14,12)(11,13,15,16)(17,19)(18,23)(20,26)(21,27)(22,24)(25,31)(28,32)(29,30)(33,36,39,38)(34,40,42,41)(35,44,46,45)(37,47,43,48);(1,5,7,14)(2,8,4,12)(3,11,9,15)(6,13,10,16)(17,20,22,23)(18,24,26,19)(21,28,30,31)(25,29,32,27)(33,37)(34,35)(36,47)(38,48)(39,43)(40,44)(41,45)(42,46);(1,6,7,10)(2,9,4,3)(5,13,14,16)(8,15,12,11)(17,21,22,30)(18,25,26,32)(19,27,24,29)(20,28,23,31)(33,38,39,36)(34,41,42,40)(35,45,46,44)(37,48,43,47)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=gh,d^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8839328);