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×#3515 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2943 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)(13,14,18,25,20,26,19,24)(15,27,31,39,32,38,30,21)(16,33,35,22,17,23,34,41)(28,40,43,37,29,36,44,42);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15,20,32)(14,21,26,39)(16,29,17,28)(18,30,19,31)(22,40,41,36)(23,42,33,37)(24,27,25,38)(34,44,35,43);(6,9)(10,12)(13,16,20,17)(14,22,26,41)(15,28,32,29)(18,34,19,35)(21,36,39,40)(23,25,33,24)(27,37,38,42)(30,43,31,44);(6,9)(10,12)(13,17,20,16)(14,23,26,33)(15,29,32,28)(18,34,19,35)(21,37,39,42)(22,25,41,24)(27,36,38,40)(30,43,31,44);(5,7,8,11)(6,10,9,12)(13,18,20,19)(14,24,26,25)(15,30,32,31)(16,34,17,35)(21,27,39,38)(22,23,41,33)(28,43,29,44)(36,37,40,42)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^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, 6398708);