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 | #43087 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4032 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(4,12)(8,15)(10,16)(13,14)(17,18,23,28)(19,27,22,24)(20,26,21,25)(29,31,30,32)(33,34,40,43)(35,44,46,41)(36,38,37,47)(39,48,45,42);(1,2)(3,7)(4,8)(5,9)(6,11)(10,14)(12,15)(13,16)(17,19,23,22)(18,24,28,27)(20,29,21,30)(25,31,26,32)(33,35)(34,41)(36,39)(37,45)(38,42)(40,46)(43,44)(47,48);(1,3)(2,6)(4,10)(5,11)(7,9)(8,13)(12,16)(14,15)(17,20,23,21)(18,25,28,26)(19,29,22,30)(24,31,27,32)(33,36,40,37)(34,38,43,47)(35,39,46,45)(41,42,44,48);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,21,23,20)(18,26,28,25)(19,29,22,30)(24,31,27,32)(33,37,40,36)(34,38,43,47)(35,45,46,39)(41,42,44,48);(1,2)(3,6)(4,8)(5,9)(7,11)(10,13)(12,15)(14,16)(17,22,23,19)(18,27,28,24)(20,30,21,29)(25,32,26,31)(33,38)(34,37)(35,42)(36,43)(39,44)(40,47)(41,45)(46,48);(1,4)(2,8)(3,10)(5,12)(6,13)(7,14)(9,15)(11,16)(17,18)(19,24)(20,25)(21,26)(22,27)(23,28)(29,31)(30,32)(33,39,40,45)(34,42,43,48)(35,36,46,37)(38,44,47,41)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,d^2=g,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8044314);