Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470185 |
| 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 |
| prime radical | C2×#3516 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2940 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)(7,11)(10,12)(13,14,17,21,19,24,18,23)(15,25,27,35,29,33,28,20)(16,22,31,36,30,37,32,38)(26,39,41,44,40,43,42,34);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15,19,29)(14,20,24,35)(16,26,30,40)(17,28,18,27)(21,33,23,25)(22,34,37,44)(31,42,32,41)(36,43,38,39);(6,9)(10,12)(14,21)(17,18)(20,33)(22,36)(23,24)(25,35)(27,28)(31,32)(34,43)(37,38)(39,44)(41,42);(6,9)(10,12)(13,16,19,30)(14,22,24,37)(15,26,29,40)(17,31,18,32)(20,34,35,44)(21,36,23,38)(25,39,33,43)(27,41,28,42);(5,7,8,11)(6,10,9,12)(16,30)(22,37)(26,40)(31,32)(34,44)(36,38)(39,43)(41,42)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=i,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6398704);