Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6443343 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2433 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4918 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,8,12)(3,6,11,7)(4,15,17,13)(5,10,20,26)(9,21,24,18)(14,22,25,19)(16,28,29,27)(23,30,32,31);(1,3,6,12,8,11,7,2)(4,13,18,9,17,15,21,24)(5,14,19,10,20,25,22,26)(16,27,30,32,29,28,31,23);(1,4,7,18,8,17,6,21)(2,9,11,15,12,24,3,13)(5,16,19,31,20,29,22,30)(10,23,25,28,26,32,14,27);(1,5)(2,10)(3,14)(4,16)(6,22)(7,19)(8,20)(9,23)(11,25)(12,26)(13,27)(15,28)(17,29)(18,31)(21,30)(24,32);(4,17)(9,24)(13,15)(16,29)(18,21)(23,32)(27,28)(30,31)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=g,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,h^a=hi,c^b=ch,d^b=dh,h^b=hi,e^c=ei,f^c=fi,f^d=fi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72213423, #72213913, #72214632, … |
| maximal normal subgroup of | #72213423, #72213913, #72214632, … |
| maximal quotient of | #72251065, #72251079, #72251179, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350540);