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 | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1856 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,8)(9,11)(10,12)(13,14,18,23,19,24,15,20)(16,27,25,35,29,37,32,21)(17,30,26,36,31,38,33,22)(28,34,42,44,41,43,39,40);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15,19,18)(14,20,24,23)(16,25,29,32)(17,26,31,33)(21,27,35,37)(22,30,36,38)(28,39,41,42)(34,40,43,44);(2,4)(5,8,9,10)(6,7,11,12)(13,16)(14,21)(15,25)(17,28)(18,32)(19,29)(20,27)(22,34)(23,37)(24,35)(26,39)(30,40)(31,41)(33,42)(36,43)(38,44);(6,11)(7,12)(13,17,19,31)(14,22,24,36)(15,26,18,33)(16,28,29,41)(20,30,23,38)(21,34,35,43)(25,39,32,42)(27,40,37,44);(16,29)(21,35)(25,32)(27,37)(28,41)(34,43)(39,42)(40,44)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=f,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,e^c=ei,g^c=gi,g^d=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7410682);