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 | C2×#555 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1814 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)(7,12)(8,10)(9,11)(13,14,20,25)(15,21,29,40)(16,30,32,42)(17,33,36,44)(18,24,19,37)(22,27,43,31)(23,26,41,35)(28,38,34,39);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15)(14,21)(16,17)(18,28)(19,34)(20,29)(22,41)(23,43)(24,38)(25,40)(26,31)(27,35)(30,33)(32,36)(37,39)(42,44);(5,8,9,7)(6,10,11,12)(13,16)(14,22)(15,26)(17,34)(18,31)(19,27)(20,32)(21,33)(23,39)(24,42)(25,43)(28,36)(29,35)(30,37)(38,41)(40,44);(2,4)(13,17,20,36)(14,23,25,41)(15,27,29,31)(16,28,32,34)(18,35,19,26)(21,30,40,42)(22,38,43,39)(24,44,37,33);(6,11)(10,12)(13,18,20,19)(14,24,25,37)(15,28,29,34)(16,31,32,27)(17,26,36,35)(21,38,40,39)(22,42,43,30)(23,33,41,44)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=f,d^2=i,e^2=i,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,d^c=di,g^c=gi,h^c=hi,e^d=ei,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, 7422979);