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 | #45091 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2047 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14,18,23)(15,24,28,20)(16,21,29,36)(17,31,19,22)(25,39,42,33)(26,35,27,34)(30,38,32,37)(40,44,41,43);(1,2)(3,4)(6,9)(10,12)(13,15,18,28)(14,20,23,24)(16,25,29,42)(17,27,19,26)(21,33,36,39)(22,34,31,35)(30,41,32,40)(37,43,38,44);(2,4)(13,16,18,29)(14,21,23,36)(15,25,28,42)(17,32,19,30)(20,33,24,39)(22,38,31,37)(26,41,27,40)(34,44,35,43);(2,4)(5,7,8,11)(6,10,9,12)(15,26)(16,29)(17,19)(20,34)(21,36)(22,31)(24,35)(25,40)(27,28)(33,43)(39,44)(41,42);(6,9)(10,12)(13,17,18,19)(14,22,23,31)(15,27,28,26)(16,30,29,32)(20,34,24,35)(21,37,36,38)(25,41,42,40)(33,43,39,44)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=i,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=ei,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7423567);