Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500434 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 | |
1820 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)(13,14)(15,20)(16,27)(17,22)(18,33)(19,23)(21,30)(24,40)(25,35)(26,36)(28,42)(29,39)(31,43)(32,37)(34,41)(38,44);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15,19,25)(14,20,23,35)(16,28,31,24)(17,32,18,26)(21,38,39,34)(22,37,33,36)(27,42,43,40)(29,41,30,44);(6,9)(7,11)(13,16,18,30,19,31,17,29)(14,21,22,27,23,39,33,43)(15,24,26,41,25,28,32,44)(20,34,37,40,35,38,36,42);(5,7,8,11)(6,10,9,12)(15,25)(20,35)(24,28)(26,32)(34,38)(36,37)(40,42)(41,44);(6,9)(10,12)(13,17,19,18)(14,22,23,33)(15,26,25,32)(16,29,31,30)(20,36,35,37)(21,27,39,43)(24,41,28,44)(34,42,38,40)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=g,d^2=hi,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dhi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6427056);