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 | |
4324 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,4)(3,7,9,11)(5,8,13,12)(10,14,15,16)(17,18,23,25)(19,24,27,32)(20,28,29,26)(21,22,30,31)(33,34)(35,38)(36,45)(37,41)(39,42)(40,44)(43,48)(46,47);(2,4)(7,11)(8,12)(14,16)(17,19)(18,24)(20,21)(22,28)(23,27)(25,32)(26,31)(29,30)(33,35)(34,38)(36,37)(39,43)(40,46)(41,45)(42,48)(44,47);(3,9)(5,13)(7,11)(8,12)(18,25)(19,27)(20,29)(22,31)(35,43)(36,44)(38,48)(40,45);(1,3,6,9)(2,7,4,11)(5,10,13,15)(8,14,12,16)(17,20,23,29)(18,26,25,28)(19,21,27,30)(22,24,31,32)(33,36)(34,40)(35,37)(38,46)(39,44)(41,48)(42,45)(43,47);(1,4,6,2)(3,7,9,11)(5,12,13,8)(10,14,15,16)(17,21)(18,22)(19,20)(23,30)(24,28)(25,31)(26,32)(27,29)(33,37)(34,41)(35,44)(36,43)(38,40)(39,47)(42,46)(45,48);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,22)(18,21)(19,28)(20,24)(23,31)(25,30)(26,27)(29,32)(33,38)(34,35)(36,46)(37,40)(39,48)(41,44)(42,43)(45,47)Polycyclic presentation:
2^9|a^2=gh,d^2=gh,e^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, 8834442);