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) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(7,14)(10,12)(15,16)(17,18,23,24)(19,25,20,26)(21,22,27,32)(28,31,30,29)(33,34)(35,44)(36,47)(37,38)(39,40)(41,45)(42,46)(43,48);(1,2)(3,7)(4,11)(5,6)(8,13)(9,14)(10,16)(12,15)(18,24)(21,27)(25,26)(28,30)(34,40)(35,45)(37,43)(42,47);(2,6)(7,14)(8,11)(15,16)(17,19,23,20)(18,25,24,26)(21,28,27,30)(22,31,32,29)(33,35)(34,41)(36,43)(37,46)(38,47)(39,45)(40,44)(42,48);(1,3)(2,7)(4,12)(5,9)(6,14)(8,16)(10,13)(11,15)(17,20,23,19)(18,26,24,25)(21,28,27,30)(22,31,32,29)(33,36,39,46)(34,42,40,47)(35,43,45,37)(38,41,48,44);(1,4)(2,8)(3,10)(5,13)(6,11)(7,15)(9,12)(14,16)(17,21,23,27)(18,22,24,32)(19,28,20,30)(25,31,26,29)(33,37,39,43)(34,38,40,48)(35,46,45,36)(41,47,44,42);(1,2)(3,7)(4,8)(5,6)(9,14)(10,15)(11,13)(12,16)(17,22)(18,27)(19,29)(20,31)(21,24)(23,32)(25,28)(26,30)(33,38)(34,43)(35,47)(36,41)(37,40)(39,48)(42,45)(44,46)Polycyclic presentation:
2^9|a^2=gh,c^2=gh,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,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, 8832228);