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 | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3998 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)(4,8)(5,6)(9,12)(10,14)(11,13)(15,16)(17,18,23,27)(19,24,29,32)(20,30,28,25)(21,26,22,31)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(2,6)(7,12)(8,13)(14,16)(17,19)(18,24)(20,21)(22,28)(23,29)(25,31)(26,30)(27,32)(33,35,38,37)(34,39,42,41)(36,44,45,43)(40,47,48,46);(1,3)(2,7)(4,10)(5,9)(6,12)(8,14)(11,15)(13,16)(17,20,23,28)(18,25,27,30)(19,21,29,22)(24,31,32,26)(33,36,38,45)(34,40,42,48)(35,43,37,44)(39,46,41,47);(2,6)(3,9)(8,13)(10,15)(17,21,23,22)(18,26,27,31)(19,28,29,20)(24,25,32,30)(33,37,38,35)(34,41,42,39)(36,43,45,44)(40,46,48,47);(2,6)(7,12)(8,13)(14,16)(17,22,23,21)(18,26,27,31)(19,28,29,20)(24,30,32,25)(34,42)(39,41)(40,48)(46,47);(1,4,5,11)(2,8,6,13)(3,10,9,15)(7,14,12,16)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=hi,d^2=hi,e^2=h,f^2=g,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, 7999984);