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 |
| prime radical | #37217 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4054 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,13,15,8)(4,16,14,10)(5,11,12,6)(17,18,22,26)(19,20,24,23)(21,25,30,32)(27,28,31,29)(33,34)(35,37)(36,44)(38,42)(39,41)(40,46)(43,47)(45,48);(1,3,7,15)(2,8,9,13)(4,12,14,5)(6,16,11,10)(17,19,22,24)(18,23,26,20)(21,27,30,31)(25,29,32,28)(33,34,38,42)(35,37,41,39)(36,40,46,44)(43,45,48,47);(2,9)(4,14)(5,12)(8,13)(17,20,22,23)(18,24,26,19)(21,29,30,28)(25,27,32,31)(33,35,38,41)(34,39,42,37)(36,43,46,48)(40,47,44,45);(1,4,7,14)(2,10,9,16)(3,12,15,5)(6,13,11,8)(17,20)(18,19)(21,28)(22,23)(24,26)(25,27)(29,30)(31,32)(33,36)(34,40)(35,43)(37,45)(38,46)(39,47)(41,48)(42,44);(1,5,7,12)(2,11,9,6)(3,14,15,4)(8,16,13,10)(17,21)(18,25)(19,27)(20,28)(22,30)(23,29)(24,31)(26,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,47)(44,46)(45,48);(1,6,7,11)(2,12,9,5)(3,10,15,16)(4,13,14,8)(17,19)(18,20)(21,27)(22,24)(23,26)(25,28)(29,32)(30,31)(33,37,38,39)(34,41,42,35)(36,45,46,47)(40,48,44,43)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=ghi,d^2=hi,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8714004);