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 | #54328 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4027 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)(3,5)(4,12)(6,7)(8,10)(9,14)(11,15)(13,16)(17,18,23,28)(19,25,20,24)(21,30,31,26)(22,27,32,29)(33,34,39,43)(35,40,44,48)(36,38,42,47)(37,46,45,41);(2,7)(3,10)(9,12)(11,16)(17,19,23,20)(18,24,28,25)(21,27,31,29)(22,26,32,30)(35,44)(37,45)(40,48)(41,46);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,13)(14,16)(17,19,23,20)(18,25,28,24)(21,29,31,27)(22,26,32,30)(33,35,39,44)(34,40,43,48)(36,45,42,37)(38,41,47,46);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,13)(14,16)(17,20,23,19)(18,25,28,24)(21,27,31,29)(22,26,32,30)(33,36)(34,38)(35,45)(37,44)(39,42)(40,41)(43,47)(46,48);(1,4,6,14)(2,9,7,12)(3,11,10,16)(5,13,8,15)(17,21)(18,26)(19,29)(20,27)(22,24)(23,31)(25,32)(28,30)(33,37,39,45)(34,41,43,46)(35,36,44,42)(38,40,47,48);(1,5,6,8)(2,3,7,10)(4,13,14,15)(9,11,12,16)(17,22,23,32)(18,27,28,29)(19,26,20,30)(21,24,31,25)(33,38)(34,42)(35,41)(36,43)(37,40)(39,47)(44,46)(45,48)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=gi,d^2=gh,e^2=i,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, 8400092);