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 | |
2000 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)(16,31,34,41)(17,36,28,44)(18,26,20,24)(22,33,39,35)(23,32,42,37)(29,40,38,43);(1,2)(3,4)(6,9)(10,12)(13,15,19,30)(14,21,25,27)(16,32,34,37)(17,35,28,33)(18,29,20,38)(22,36,39,44)(23,41,42,31)(24,40,26,43);(13,16,19,34)(14,22,25,39)(15,28,30,17)(18,35,20,33)(21,23,27,42)(24,31,26,41)(29,37,38,32)(36,40,44,43);(2,4)(5,7,8,11)(6,10,9,12)(13,17,19,28)(14,23,25,42)(15,16,30,34)(18,37,20,32)(21,39,27,22)(24,36,26,44)(29,33,38,35)(31,43,41,40);(6,9)(10,12)(13,18,19,20)(14,24,25,26)(15,29,30,38)(16,33,34,35)(17,37,28,32)(21,40,27,43)(22,41,39,31)(23,36,42,44)Polycyclic presentation:
2^9|a^2=i,b^2=i,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, 7424109);