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 | #36905 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3947 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,11)(3,8,12,16)(4,6,10,15)(5,14,13,9)(18,24)(20,28)(21,29)(25,32)(33,34,38,39)(35,37,42,40)(36,44,46,41)(43,47,48,45);(3,12)(5,13)(8,16)(9,14)(17,18)(19,25)(20,26)(21,30)(22,28)(23,24)(27,29)(31,32)(34,39)(35,42)(41,44)(43,48);(1,3,7,12)(2,8,11,16)(4,13,10,5)(6,9,15,14)(17,19,23,31)(18,25,24,32)(20,29,28,21)(22,30,26,27)(33,35,38,42)(34,40,39,37)(36,43,46,48)(41,47,44,45);(1,4)(2,6)(3,13)(5,12)(7,10)(8,9)(11,15)(14,16)(17,20)(18,26)(19,29)(21,31)(22,24)(23,28)(25,27)(30,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,46)(45,48);(1,5,7,13)(2,9,11,14)(3,4,12,10)(6,8,15,16)(17,21,23,29)(18,27,24,30)(19,20,31,28)(22,32,26,25)(33,36)(34,41)(35,43)(37,45)(38,46)(39,44)(40,47)(42,48);(1,6)(2,10)(3,9)(4,11)(5,8)(7,15)(12,14)(13,16)(17,22,23,26)(18,28,24,20)(19,30,31,27)(21,32,29,25)(33,37,38,40)(34,42,39,35)(36,45,46,47)(41,48,44,43)Polycyclic presentation:
2^9|a^2=i,c^2=ghi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8399983);