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 | |
4116 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,8,4,11)(5,9,13,14)(10,15,12,16)(17,18,22,26)(19,20,24,23)(21,29,30,25)(27,31,32,28)(33,34)(35,40)(36,38)(37,44)(39,43)(41,45)(42,46)(47,48);(2,7)(8,11)(9,14)(15,16)(33,35)(34,39)(36,40)(37,42)(38,43)(41,46)(44,47)(45,48);(1,3,6,4)(2,8,7,11)(5,10,13,12)(9,15,14,16)(17,19)(18,23)(20,26)(21,27)(22,24)(25,31)(28,29)(30,32)(33,34,38,36)(35,40,43,39)(37,41,45,44)(42,47,48,46);(1,4,6,3)(2,8,7,11)(5,10,13,12)(9,16,14,15)(17,19)(18,20)(21,27)(22,24)(23,26)(25,28)(29,31)(30,32)(33,36,38,34)(35,40,43,39)(37,44,45,41)(42,47,48,46);(1,2)(3,8)(4,11)(5,9)(6,7)(10,15)(12,16)(13,14)(17,20)(18,24)(19,26)(21,28)(22,23)(25,32)(27,29)(30,31)(33,35)(34,40)(36,39)(37,42)(38,43)(41,47)(44,46)(45,48);(1,5,6,13)(2,9,7,14)(3,10,4,12)(8,15,11,16)(17,21,22,30)(18,25,26,29)(19,27,24,32)(20,28,23,31)(33,37)(34,41)(35,42)(36,44)(38,45)(39,46)(40,47)(43,48)Polycyclic presentation:
2^9|a^2=gh,c^2=g,d^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10317190);