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×#548 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1828 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)(15,21,29,42)(16,30,33,39)(17,34,31,38)(18,24,19,36)(22,32,43,27)(23,37,44,26)(28,40,35,41);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15)(14,21)(16,31)(17,33)(18,28)(19,35)(20,29)(22,23)(24,40)(25,42)(26,27)(30,38)(32,37)(34,39)(36,41)(43,44);(13,16,20,33)(14,22,25,43)(15,26,29,37)(17,35,31,28)(18,27,19,32)(21,38,42,34)(23,41,44,40)(24,39,36,30);(2,4)(5,8,9,7)(6,10,11,12)(13,17)(14,23)(15,27)(16,28)(18,26)(19,37)(20,31)(21,39)(22,40)(24,38)(25,44)(29,32)(30,42)(33,35)(34,36)(41,43);(6,11)(10,12)(13,18,20,19)(14,24,25,36)(15,28,29,35)(16,32,33,27)(17,26,31,37)(21,40,42,41)(22,30,43,39)(23,38,44,34)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=i,d^2=f,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,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, 7425189);