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 | #53061 | C2 |
| 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:
(1,2,6,9)(3,10,5,7)(4,8,12,15)(11,16,13,14)(17,18,21,24)(19,25,27,22)(20,28,29,23)(26,30,32,31)(33,34)(35,39)(36,40)(37,45)(38,41)(42,46)(43,47)(44,48);(1,3,6,5)(2,7,9,10)(4,11,12,13)(8,14,15,16)(17,19,21,27)(18,22,24,25)(20,26,29,32)(23,30,28,31)(33,35,38,37)(34,39,41,45)(36,43,44,42)(40,47,48,46);(1,4,6,12)(2,8,9,15)(3,11,5,13)(7,14,10,16)(17,20,21,29)(18,23,24,28)(19,26,27,32)(22,30,25,31)(33,36)(34,40)(35,42)(37,43)(38,44)(39,46)(41,48)(45,47);(2,9)(4,12)(7,10)(11,13)(18,24)(19,27)(23,28)(26,32)(36,44)(40,48)(42,43)(46,47);(2,9)(7,10)(8,15)(14,16)(33,37,38,35)(34,39,41,45)(36,43,44,42)(40,46,48,47);(1,5,6,3)(2,7,9,10)(4,13,12,11)(8,14,15,16)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^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, 8000249);