Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518920 |
| 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 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2155 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,11)(8,9)(10,12)(13,14,19,24)(15,26,28,21)(16,29,32,38)(17,34,35,40)(18,25,20,23)(22,31,39,33)(27,42,43,37)(30,41,44,36);(1,2)(3,4)(6,9)(10,11)(13,15,19,28)(14,21,24,26)(16,30,32,44)(17,20,35,18)(22,37,39,42)(23,34,25,40)(27,31,43,33)(29,36,38,41);(13,16,18,31,19,32,20,33)(14,22,23,38,24,39,25,29)(15,27,35,44,28,43,17,30)(21,36,40,42,26,41,34,37);(2,4)(13,17,20,28,19,35,18,15)(14,21,25,34,24,26,23,40)(16,30,33,43,32,44,31,27)(22,36,29,37,39,41,38,42);(5,7,8,12)(6,10,9,11)(13,18,19,20)(14,23,24,25)(15,17,28,35)(16,31,32,33)(21,34,26,40)(22,38,39,29)(27,30,43,44)(36,37,41,42)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=gi,d^2=g,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7469696);