Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437537 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3132 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14,19,22)(15,25,28,35)(16,21,33,38)(17,24,18,23)(20,29,36,26)(27,37,41,43)(30,34,42,44)(31,40,32,39);(2,4)(13,15)(14,20)(16,30)(17,29)(18,26)(19,28)(21,37)(22,36)(23,35)(24,25)(27,32)(31,41)(33,42)(34,40)(38,43)(39,44);(6,9)(10,12)(15,26)(17,18)(20,25)(23,24)(27,30)(28,29)(31,32)(34,37)(35,36)(39,40)(41,42)(43,44);(13,16,19,33)(14,21,22,38)(15,27,28,41)(17,32,18,31)(20,34,36,44)(23,40,24,39)(25,37,35,43)(26,30,29,42);(5,7,8,11)(6,10,9,12)(14,22)(15,28)(21,38)(23,24)(26,29)(27,41)(30,42)(39,40)Polycyclic presentation:
2^9|a^2=i,d^2=i,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,h^a=hi,c^b=ch,d^b=dh,e^b=ei,f^b=fi,h^b=hi,f^c=fi,h^c=hi,f^d=fi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6361206);