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 | #54036 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3769 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18,23,21)(19,26,24,20)(22,29,31,25)(27,28,32,30)(33,34)(35,40)(36,44)(37,42)(38,39)(41,46)(43,45)(47,48);(3,9)(7,13)(10,15)(14,16)(17,19)(18,20)(21,26)(22,27)(23,24)(25,28)(29,30)(31,32)(34,39)(35,42)(41,44)(43,48);(2,6)(7,13)(8,11)(14,16)(17,20,23,26)(18,24,21,19)(22,30,31,28)(25,27,29,32)(33,35,38,42)(34,40,39,37)(36,43,46,48)(41,47,44,45);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,20)(18,19)(21,24)(22,28)(23,26)(25,27)(29,32)(30,31)(33,36,38,46)(34,41,39,44)(35,43,42,48)(37,45,40,47);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,21,23,18)(19,26,24,20)(22,29,31,25)(27,30,32,28);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,22,23,31)(18,25,21,29)(19,27,24,32)(20,28,26,30)(33,37,38,40)(34,42,39,35)(36,45,46,47)(41,48,44,43)Polycyclic presentation:
2^9|a^2=gi,c^2=g,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^c=fgi,e^d=egi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9388656);