Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| 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 | #30502 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1852 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)(7,8)(9,11)(10,12)(13,14,18,24)(15,25,28,20)(16,29,31,37)(17,23,19,21)(22,32,38,30)(26,40,42,44)(27,36,33,34)(35,43,39,41);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15,18,28)(14,20,24,25)(16,26,31,42)(17,33,19,27)(21,34,23,36)(22,35,38,39)(29,44,37,40)(30,43,32,41);(2,4)(14,21)(17,19)(20,34)(22,37)(23,24)(25,36)(27,33)(29,38)(30,32)(35,40)(39,44)(41,43);(6,11)(7,12)(13,16,18,31)(14,22,24,38)(15,26,28,42)(17,32,19,30)(20,35,25,39)(21,37,23,29)(27,43,33,41)(34,40,36,44);(5,8,9,10)(6,7,11,12)(13,17,18,19)(14,23,24,21)(15,27,28,33)(16,30,31,32)(20,34,25,36)(22,37,38,29)(26,41,42,43)(35,44,39,40)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dfg,g^a=gi,c^b=ch,d^b=dfi,e^b=ei,e^c=ei,g^c=gi,e^d=ei,g^d=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7410269);