Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437986 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2410 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4538 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,11,7,9,6,12)(3,13,14,24,16,22,17,8)(4,10,19,27,20,25,21,28)(15,26,18,31,29,32,30,23);(1,3)(2,8)(4,18)(5,17)(6,14)(7,16)(9,24)(10,26)(11,22)(12,13)(15,19)(20,30)(21,29)(23,27)(25,32)(28,31);(2,9)(3,14)(5,6)(8,22)(10,25)(13,24)(15,18)(16,17)(19,21)(23,32)(26,31)(29,30);(1,4)(2,10)(3,15)(5,19)(6,21)(7,20)(8,23)(9,25)(11,27)(12,28)(13,26)(14,18)(16,29)(17,30)(22,32)(24,31);(2,9)(8,24)(10,25)(11,12)(13,22)(23,31)(26,32)(27,28)Polycyclic presentation:
2^9|a^2=f,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,h^b=hi,f^c=fi,h^c=hi,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72215784, #72215938, #72219692, … |
| maximal normal subgroup of | #72215784, #72215938, #72219692, … |
| maximal quotient of | #72312228, C2×#6437986 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345182);