Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573615 |
| 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 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2142 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)(13,14,18,23)(15,25,27,20)(16,30,32,38)(17,24,19,22)(21,33,37,31)(26,40,41,44)(28,35,29,36)(34,42,39,43);(1,2)(3,4)(6,9)(10,12)(13,15)(14,20)(16,26)(17,28)(18,27)(19,29)(21,34)(22,35)(23,25)(24,36)(30,44)(31,42)(32,41)(33,43)(37,39)(38,40);(2,4)(13,16)(14,21)(15,26)(17,31)(18,32)(19,33)(20,34)(22,30)(23,37)(24,38)(25,39)(27,41)(28,42)(29,43)(35,44)(36,40);(1,2)(3,4)(5,7,8,11)(6,10,9,12)(15,27)(20,25)(26,41)(28,29)(34,39)(35,36)(40,44)(42,43);(2,4)(13,17,18,19)(14,22,23,24)(15,28,27,29)(16,31,32,33)(20,35,25,36)(21,30,37,38)(26,42,41,43)(34,44,39,40)Polycyclic presentation:
2^9|a^2=i,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6482897);