Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470185 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #37379 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3098 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)(13,14,20,24)(15,27,30,22)(16,31,29,21)(17,23,34,42)(18,26,19,25)(28,43,33,38)(32,39,44,37)(35,41,36,40);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15,20,30)(14,21,24,31)(16,19,29,18)(17,33,34,28)(22,26,27,25)(23,39,42,37)(32,35,44,36)(38,40,43,41);(6,9)(10,12)(13,16,20,29)(14,22,24,27)(15,18,30,19)(17,32,34,44)(21,25,31,26)(23,38,42,43)(28,35,33,36)(37,40,39,41);(6,9)(10,12)(13,17)(14,23)(15,28)(16,32)(18,35)(19,36)(20,34)(21,37)(22,38)(24,42)(25,40)(26,41)(27,43)(29,44)(30,33)(31,39);(5,7,8,11)(6,10,9,12)(14,24)(17,34)(21,31)(22,27)(25,26)(28,33)(32,44)(35,36)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=i,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,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, 6398339);