Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500908 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1760 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,4)(5,7)(6,8)(9,10,14,19,15,21,13,20)(11,16,23,18,25,33,24,32)(12,26,28,17,30,35,29,36)(22,34,38,31,37,40,27,39);(1,3)(2,6)(4,7)(5,8)(9,11)(10,16)(12,27)(13,24)(14,23)(15,25)(17,34)(18,19)(20,32)(21,33)(22,28)(26,39)(29,37)(30,38)(31,35)(36,40);(1,4,5,6)(2,3,7,8)(9,12)(10,17)(11,22)(13,28)(14,29)(15,30)(16,31)(18,34)(19,26)(20,35)(21,36)(23,27)(24,38)(25,37)(32,40)(33,39);(2,7)(3,8)(9,11)(10,18)(12,22)(13,23)(14,24)(15,25)(16,19)(17,34)(20,33)(21,32)(26,31)(27,28)(29,38)(30,37)(35,39)(36,40);(9,13,15,14)(10,19,21,20)(11,23,25,24)(12,28,30,29)(16,32,33,18)(17,26,36,35)(22,27,37,38)(31,40,39,34)Polycyclic presentation:
2^9|a^2=gi,c^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,e^b=ei,h^b=hi,g^c=gi,h^c=hi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7410666);