Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| 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 | #45243 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1869 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)(8,10)(9,12)(13,14,18,20)(15,25,29,37)(16,30,32,38)(17,24,19,23)(21,26,34,28)(22,31,35,33)(27,36,42,43)(39,41,44,40);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(14,20)(15,26)(16,31)(17,19)(21,25)(22,30)(28,29)(32,33)(34,37)(35,38)(36,43)(40,41);(5,7,9,8)(6,11,12,10)(13,15,18,29)(14,21,20,34)(16,27,32,42)(17,26,19,28)(22,36,35,43)(23,37,24,25)(30,39,38,44)(31,40,33,41);(2,4)(13,16,18,32)(14,22,20,35)(15,27,29,42)(17,31,19,33)(21,36,34,43)(23,38,24,30)(25,39,37,44)(26,40,28,41);(5,8,9,7)(6,11,12,10)(13,17,18,19)(14,23,20,24)(15,28,29,26)(16,31,32,33)(21,25,34,37)(22,38,35,30)(27,41,42,40)(36,39,43,44)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=f,d^2=i,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=cgi,d^b=dh,e^b=ei,g^b=gi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,g^d=gi,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7427729);