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 | #2670 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4052 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,7,9,10)(4,12,13,8)(11,16,15,14)(17,18)(19,24)(20,26)(21,27)(22,30)(23,25)(28,32)(29,31)(33,34,38,40)(35,41,42,36)(37,39,45,46)(43,47,48,44);(2,6)(7,10)(8,12)(14,16)(17,19)(18,24)(20,23)(21,28)(22,29)(25,26)(27,32)(30,31)(33,34)(35,41)(36,42)(37,39)(38,40)(43,47)(44,48)(45,46);(2,6)(3,9)(4,13)(14,16)(18,25)(19,20)(21,31)(22,32)(35,42)(36,41)(37,45)(39,46);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,20,23,19)(18,26,25,24)(21,28,31,30)(22,27,32,29)(33,35,38,42)(34,36,40,41)(37,43,45,48)(39,44,46,47);(1,3,5,9)(2,7,6,10)(4,11,13,15)(8,14,12,16)(17,21,23,31)(18,27,25,29)(19,28,20,30)(22,24,32,26)(33,36)(34,35)(37,44)(38,41)(39,43)(40,42)(45,47)(46,48);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,22)(18,28)(19,29)(20,27)(21,24)(23,32)(25,30)(26,31)(33,37)(34,39)(35,43)(36,44)(38,45)(40,46)(41,47)(42,48)Polycyclic presentation:
2^9|a^2=g,d^2=gh,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8966268);