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 | #42746 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3944 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(17,18,23,24)(19,27,21,25)(20,29,30,26)(22,28,32,31)(33,34,38,41)(35,39,43,37)(36,44,46,40)(42,45,48,47);(1,2)(3,8)(4,9)(5,10)(6,7)(11,15)(12,13)(14,16)(18,24)(20,30)(25,27)(28,31)(33,35,38,43)(34,37,41,39)(36,42,46,48)(40,45,44,47);(2,7)(5,15)(8,13)(9,16)(17,19,23,21)(18,25,24,27)(20,28,30,31)(22,26,32,29)(33,35)(34,39)(36,42)(37,41)(38,43)(40,47)(44,45)(46,48);(1,3)(2,8)(4,10)(5,9)(6,12)(7,13)(11,14)(15,16)(17,20)(18,26)(19,28)(21,31)(22,27)(23,30)(24,29)(25,32)(33,36,38,46)(34,40,41,44)(35,42,43,48)(37,45,39,47);(1,4)(2,9)(3,11)(5,13)(6,14)(7,16)(8,15)(10,12)(17,21,23,19)(18,27,24,25)(20,28,30,31)(22,26,32,29)(33,37,38,39)(34,35,41,43)(36,45,46,47)(40,42,44,48);(1,5,6,15)(2,10,7,11)(3,9,12,16)(4,13,14,8)(17,22,23,32)(18,28,24,31)(19,29,21,26)(20,27,30,25)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=h,d^2=gh,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8229320);