Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #2619 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4069 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)(3,10)(4,8)(5,7)(6,9)(11,15)(12,14)(13,16)(17,18)(19,25)(20,28)(21,23)(22,27)(24,30)(26,31)(29,32)(33,34)(35,39)(36,40)(37,45)(38,41)(42,46)(43,47)(44,48);(1,3,6,5)(2,7,9,10)(4,11,13,12)(8,14,16,15)(17,19,21,27)(18,22,23,25)(20,29,30,26)(24,32,28,31)(33,35,38,37)(34,39,41,45)(36,43,44,42)(40,47,48,46);(1,4)(2,8)(3,11)(5,12)(6,13)(7,14)(9,16)(10,15)(18,23)(22,25)(24,28)(31,32)(33,36)(34,40)(35,42)(37,43)(38,44)(39,46)(41,48)(45,47);(1,3,6,5)(2,7,9,10)(4,12,13,11)(8,15,16,14)(17,20,21,30)(18,24,23,28)(19,26,27,29)(22,31,25,32)(36,44)(40,48)(42,43)(46,47);(2,9)(7,10)(8,16)(14,15)(33,37,38,35)(34,39,41,45)(36,43,44,42)(40,46,48,47);(1,5,6,3)(2,7,9,10)(4,12,13,11)(8,14,16,15)Polycyclic presentation:
2^9|b^2=gi,d^2=g,e^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,d^c=dghi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8000250);