Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61643 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #60366 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#408 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2614 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)(3,7)(4,6)(5,8)(9,10,14,19,15,12,23,18)(11,16,21,30,22,24,35,29)(13,17,27,32,26,25,36,31)(20,28,34,39,33,37,40,38);(1,3)(2,5)(4,8)(6,7)(9,11,14,21,15,22,23,35)(10,16,19,30,12,24,18,29)(13,20,27,34,26,33,36,40)(17,28,32,39,25,37,31,38);(9,12)(10,15)(11,16)(13,25)(14,18)(17,26)(19,23)(20,28)(21,30)(22,24)(27,31)(29,35)(32,36)(33,37)(34,39)(38,40);(3,8)(5,7)(9,13)(10,17)(11,20)(12,25)(14,27)(15,26)(16,28)(18,31)(19,32)(21,34)(22,33)(23,36)(24,37)(29,38)(30,39)(35,40);(2,6)(5,7)(13,26)(17,25)(20,33)(27,36)(28,37)(31,32)(34,40)(38,39)Polycyclic presentation:
2^9|a^2=h,b^2=h,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ci,d^b=df,g^c=gi,e^d=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72314358, #72315260, #72315580, … |
| maximal normal subgroup of | #72314358, #72315260, #72315580, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6294889);