Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #452530 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16051 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
942 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,13)(3,14)(4,19)(5,11,6,12)(8,17)(9,21)(10,23)(15,24,31,29)(16,26)(18,30)(20,25,32,27)(22,28);(1,3,6,16,7,18,5,17)(2,8,12,14,13,26,11,30)(4,20,22,31,23,32,21,15)(9,27,19,24,28,25,10,29);(1,4)(2,9)(3,15,18,31)(5,22)(6,21)(7,23)(8,24,26,29)(10,12)(11,19)(13,28)(14,27,30,25)(16,32,17,20);(1,4)(2,10)(3,15,18,31)(5,21)(6,22)(7,23)(8,25,26,27)(9,12)(11,28)(13,19)(14,29,30,24)(16,20,17,32)Polycyclic presentation:
2^9|a^2=egh,b^2=f,c^2=ef,d^2=ef,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,e^c=ei,f^c=fi,g^d=gi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72251689, #72251691, #72251710, … |
| maximal normal subgroup of | #72251689, #72251691, #72251710, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 359726);