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 | #2688 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3881 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)(3,11,14,8)(4,5,10,9)(12,13,16,15)(17,18,23,27)(19,24,28,32)(20,22,26,31)(21,30,29,25)(33,34)(35,42)(36,39)(37,44)(38,41)(40,46)(43,45)(47,48);(2,7)(4,10)(8,11)(12,16)(19,28)(21,29)(24,32)(25,30)(34,36)(35,41)(40,44)(43,47);(17,19,23,28)(18,24,27,32)(20,29,26,21)(22,25,31,30)(33,35)(34,38)(36,42)(37,43)(39,41)(40,45)(44,48)(46,47);(1,3,6,14)(2,8,7,11)(4,12,10,16)(5,15,9,13)(17,20)(18,22)(19,29)(21,28)(23,26)(24,25)(27,31)(30,32)(33,36,39,34)(35,42,41,38)(37,44,46,40)(43,48,47,45);(1,4,6,10)(2,9,7,5)(3,12,14,16)(8,13,11,15)(17,21,23,29)(18,25,27,30)(19,20,28,26)(22,24,31,32)(33,37)(34,40)(35,43)(36,44)(38,45)(39,46)(41,47)(42,48);(1,5,6,9)(2,10,7,4)(3,13,14,15)(8,12,11,16)(17,22)(18,26)(19,25)(20,27)(21,24)(23,31)(28,30)(29,32)(33,38,39,42)(34,41,36,35)(37,45,46,48)(40,47,44,43)Polycyclic presentation:
2^9|a^2=gi,c^2=hi,d^2=g,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, 8399988);