Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #28152 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1583 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16);(2,6)(7,8)(11,16)(13,15)(17,18,21,25)(19,26,27,22)(20,23,29,24)(28,31,32,30);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,13,14,15)(17,19,21,27)(18,22,25,26)(20,28,29,32)(23,30,24,31);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(18,23)(20,29)(22,30)(24,25)(26,31)(28,32);(1,4,5,3)(2,7,6,8)(19,27)(22,26)(28,32)(30,31)Polycyclic presentation:
2^9|b^2=i,c^2=fgi,d^2=f,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dh,h^b=hi,e^c=ei,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72257398, #72257535, #72257667 |
| maximal normal subgroup of | #72257398, #72257535, #72257667 |
| maximal quotient of | #72308828 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7442645);