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×#547 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1954 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)(7,10)(8,9)(11,12)(13,14,19,25)(15,27,30,21)(16,31,28,44)(17,34,37,41)(18,26,20,24)(22,35,42,33)(23,36,39,32)(29,40,38,43);(1,2)(3,4)(6,9)(10,12)(13,15,19,30)(14,21,25,27)(16,32,28,36)(17,35,37,33)(18,29,20,38)(22,41,42,34)(23,31,39,44)(24,40,26,43);(5,7,8,11)(6,10,9,12)(13,16)(14,22)(15,17)(18,33)(19,28)(20,35)(21,39)(23,27)(24,31)(25,42)(26,44)(29,36)(30,37)(32,38)(34,40)(41,43);(2,4)(13,17,19,37)(14,23,25,39)(15,28,30,16)(18,32,20,36)(21,22,27,42)(24,34,26,41)(29,33,38,35)(31,40,44,43);(6,9)(10,12)(13,18,19,20)(14,24,25,26)(15,29,30,38)(16,33,28,35)(17,36,37,32)(21,40,27,43)(22,31,42,44)(23,41,39,34)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fi,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,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, 7422183);